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Consensus-Based Distributed Multiple Model UKF for Jump Markov Nonlinear Systems Wenling Li and Yingmin Jia

Abstract—This note studies the problem of distributed estimation for jump Markov nonlinear systems (JMNLSs) in a not fully connected sensor network. Based on the consensus theory, a distributed unscented Kalman filter (UKF) is first derived for nonlinear systems without jumping parameters and then it is extended to develop a distributed multiple model UKF for JMNLSs. The proposed filtering algorithm is illustrated via a simulation example involving tracking a maneuvering target. Index Terms—Consensus theory, distributed estimation, jump Markov nonlinear system (JMNLS), unscented Kalman filter (UKF).

I. INTRODUCTION Distributed estimation has attracted increasing attention with the development of large-scale sensor networks. Compared with the traditional centralized and decentralized estimation schemes, several advantages emerge such as scalability, low communication load, fast implementation and more robustness to sensor failures [1]–[3]. This is partly due to the main features of distributed algorithms, i.e, each sensor node only communicates with its neighboring peers and no fusion center is present. Based on the Kalman filter, many distributed estimation algorithms have recently been proposed, see [4]–[10] and references therein. In particular, by using dynamic average-consensus strategies to the weighted measurements and the inverse-covariance matrices, the distributed Kalman filter (DKF) has been developed in [4], [5]. In [6], consensus on estimates is used together with Kalman filters. By assuming the transfer matrices to be sparse and localized, an efficient distributed algorithm has been proposed in [7] while [8] proposes DKF and smoother by utilizing the diffusion strategy. In [9], a distributed Kriged Kalman filter has been presented for spatial estimation. Based on the concept of moving horizon estimation, a distributed algorithm for linear constrained systems has been presented in [10]. Recently, many filtering algorithms for jump Markov systems have been developed [11]–[13], of which the interacting multiple model (IMM) estimator has been shown to be one of the most cost-effective approaches. As mentioned in [14], a technical challenge is encountered when distributed fusion is applied in the multiple model environment. In fact, to generate fused estimates at the center unit at each time step, the following quantities are needed: i) the locally propagated estimates for each node, ii) the updated estimates for each node, and iii) the propagated fused estimates [14]. If the mode-dependent system dynamics models are different, then the propagated fused estimates do not exist due to the lack of a global model. Based on the IMM approach, a distributed fusion algorithm is developed for jump Markov linear systems in [14]. This distributed nature is achieved by transmitting local state estimates to a centralized unit. To our knowledge, distributed estimation algorithms have not been proposed Manuscript received January 15, 2011; revised June 05, 2011; accepted July 01, 2011. Date of publication July 14, 2011; date of current version December 29, 2011. This work was supported by the National 973 Program (2012CB821200) and the NSFC 61134005, 60921001, 90916024, 91116016). Recommended by Associate Editor A. Chiuso The authors are with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2161838

Fig. 1. Maneuvering target tracking using multiple UAVs.

yet for jump Markov systems. For example, we consider a multi-static radar tracking problem as shown in Fig. 1. On one hand, the target trajectory is composed of lines in which the target moves straight and circular arcs in which the target maneuvers with different coordinate turn rates. To accurately represent the behavior of the target over all of the motion regimes, the multiple model approach has been found to be quite effective [15]. In this approach various kinematic models are used to describe distinct motions and the target maneuvers are modeled by switching among these models governed by a Markov chain. Then the target dynamics can be represented by a jump Markov system. On the other hand, the radar emits signal that bounces off the target and is received by a team of unmanned aerial vehicles (UAVs), and each UAV makes a local time-delay and Doppler measurement. In [16], a centralized algorithm is proposed for estimating the target’s state using the single motion model approach, while it is well-known that failure of the centralized unit is vital to the entire network. In this note, rather than using a centralized processor but assuming that each UAV can only communicate with its neighbors, the distributed estimation algorithm is developed for tracking a maneuvering target in the multi-model framework. The focus of this note is on distributed estimation for discrete-time jump Markov nonlinear systems (JMNLSs) where sensor nodes only communicate with their neighbors and no fusion center is present. With the help of the IMM approach, a distributed multiple model unscented Kalman filter (UKF) is proposed based on the consensus theory and the resulting filter requires no global models in the multiple model fusion setup. To this end, two difficulties have to be overcome. First, by employing the statistical linear error propagation methodology, a pseudo measurement matrix is introduced to develop the distributed UKF. Secondly, the mode probability is computed in a distributed manner using the natural logarithm transformation, which transforms a product term into a sum one so that the consensus filter can be used directly. Simulation results show that the computation and communication costs required by each sensor to keep the constant performance increase as the number of nodes increases. Specifically, as the required iteration step

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Tc to reach consensus is quadratic in the number of nodes N and their costs are proportional to NNb Tc (Nb is the number of neighbors), it

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calculated by

Consider the following discrete-time jump Markov nonlinear system: (1) (2)

where xk 2 n and zki 2 p are the state and the ith measurement vectors, respectively. The process noise wk01 (rk ) and the measurement noise vki (rk ) are assumed to be mutually uncorrelated zero-mean white Gaussian processes with covariance matrices Qk01 (rk ) and Rki (rk ), respectively. N is the number of sensor nodes. The communication topology between nodes is described by the directed graph G = (V ; E ) where V = f1; 2; . . . ; N g denotes the node set. An edge (j; i) 2 E models that node j can transmit information to node i. The set of neighbors connected a certain node i is called the neighborhood of node i and is denoted by i = fj j(j; i) 2 Eg. f is the system dynamics function and hi is the measurement function of the ith sensor node. rk denotes the system mode which is described by a discrete-time homogenous Markov chain. It is assumed that rk takes value in a finite set M = f1; 2; . . . ; M g with transition probability matrix 5 = [rsM]M 2M , where rs Prfrk+1 = sjrk = rg for all r; s 2 M and s=1 rs = 1 for any r 2 M. For simplicity of notation, we denote f (xk01 ; rk ) = fr (xk ), hi (xk ; rk ) = hri (xk ), Qk01 (rk ) = Qr;k01 i and Rki (rk ) = Rr;k for rk = r 2 M. The aim of this note is to design a distributed filtering algorithm for the jump Markov nonlinear system in a not fully connected sensor network. The main challenge is to obtain a state estimate for each sensor node that is as accurate as that of the centralized filter. To be specific, the performance of the distributed filter is expected to be comparable to that of the IMM-UKF where the measurements from all sensor nodes are augmented as a centralized measurement.

s=0 2n

Pkijk01 =

II. PROBLEM FORMULATION

xk = f (xk01 ; rk ) + wk01 (rk ) zki = hi (xk ; rk ) + vki (rk ) i = 1; 2; . . . ; N

2n

x^ikjk01 =

follows that their costs are cubic in the number of nodes.

s=0

(6)

n where the weighted sigma points fWs ; ki;s01jk01 gs2=0 are generated according to

ki;00 1jk01 = x^ik01jk01 ; W0 =

 n+

(7)

(n + )Pki01jk01

ki;s01jk01 = x^ik01jk01 +

1 2(n + ) ; s = 1; . . . ; n ki;s0+1jnk01 = x^ik01jk01 0 (n + )Pki01jk01 1 Ws+n = 2(n + ) ; s = 1; . . . ; n Ws =

with  being a scaling factor.

(n + )Pki01jk01

s

(8)

s

(9)

denotes ei-

ther the sth row or the sth column of the matrix square root of (n + )Pki01jk01 . In general if the matrix square root A of P is of the form P = AT A, then the sigma points are formed from the rows of A. Otherwise, the columns of A are used if P = AAT [17]. Recall that the derivation of the updated equations in the DKF is based on the linear measurement matrix, while there is not an explicit expression for this solution in the unscented transform framework. To circumvent this problem, by using the statistical linear error propagation methodology, we can define a pseudo measurement matrix for each sensor node as in [18] s

i [Pxz;k ]T [Pkijk01 ]01

Hki

z^ki jk01 = i Pxz;k =

A. Distributed UKF Consider the following discrete-time nonlinear system without jumping parameters:

2n s=0 2n s=0

Ws hi (f (ki;s01jk01 ))

(10)

(11)

Ws [f (ki;s01jk01 ) 0 x^ikjk01 ]

2 [hi (f (ki;s01jk01 )) 0 z^ki jk01 ]T :

(12)

Then, the updated mean and covariance of the centralized UKF can be obtained by

(3)

x^ikjk = x^ikjk01

(4)

where xk 2 and z 2 are the state and the ith measurement vectors, respectively. The notations f , hi , wk01 and vki are defined as in (1)–(2) but they are not dependent on rk . Rather than constructing fused measurements and covariance of a centralized Kalman filter, consensus on measurements and covariance information is used to develop the DKF in [4], [5]. Then the DKF can be reduced to two dynamic consensus problems which can be resolved by developing consensus filters. This idea motives us to develop a distributed filter in the unscented transform framework. Based on the state ^ik01jk01 and covariance Pki01jk01 obtained for the ith sensor estimate x node at time step k 0 1, the predicted mean and covariance can be i k

Ws [f (ki;s01jk01 ) 0 x^ikjk01 ]

2 [f (ki;s01jk01) 0 x^ikjk01 ]T + Qk01

In this section, the distributed UKF is derived based on the consensus theory and then it is extended to develop a distributed filtering algorithm for JMNLSs by using the IMM approach.

n

(5)

i can be computed by where the cross-correlation covariance Pxz;k

III. DISTRIBUTED MULTIPLE MODEL UKF FOR JMNLSS

xk = f (xk01 ) + wk01 zki = hi (xk ) + vki ; i = 1; . . . ; N

Ws f (ki;s01jk01 )

+ Pkijk

p

N

j =1

j [Pkjjk01]01 Pxz;k [Rkj ]01 (zkj 0 z^kj jk01 ) (13)

[Pkijk ]01 = [Pkijk01 ]01 +

N j =1

j [Pkjjk01]01 Pxz;k [Rkj ]01

j 2 [Pxz;k ]T [Pkjjk01]01 :

(14)

For the convenience of further development, we define

ykjk

1

N

N j =1

j [Pkjjk01]01 Pxz;k [Rkj ]01 (zkj 0 z^kj jk01 )

(15)

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Ykjk

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1

N

N j =1

j j [Pkjjk01 ]01 Pxz;k [Rkj ]01 [Pxz;k ]T [Pkjjk01 ]01 :

(16)

The filtering algorithm can be carried out in a distributed manner if the averages (15) and (16) can be obtained by every sensor node. To this end, two consensus filters can be applied. The consensus filter is a dynamic version of the average-consensus algorithm that allows the nodes to track the averages in a not fully connected network. A simple scheme is to update the state i of each node according to some weighted linear combination of its neighbors’ states [19]

i ( + 1) = i ( ) +

N j =1

ij ( )(j ( ) 0 i ( ))

(17)

where  denotes the consensus iterating step, and ij ( ) is the linear i [Ri ]01 (z i 0 weight on j at node i. Treating yki jk [Pkijk01 ]01 Pxz;k k k i i 01 i i i 01 i z^kjk01 ) and Ykjk [Pkjk01 ] Pxz;k [Rk ] [Pxz;k ]T [Pkijk01 ]01 as input states of two consensus filters (17), the outputs y^ki jk and Y^kijk can asymptotically track the values of ykjk and Ykjk , respectively. Therefore, instead of using the centralized update in (13)–(14), the distributed estimates can be written as

x^ikjk = x^ikjk01 + NPkijk y^ki jk [Pkijk ]01 = [Pkijk01 ]01 + N Y^kijk :

(18) (19)

Remark 1: For the distributed UKF, consensus on weighted measurement yki jk and covariance information Ykijk has been used together with the UKF and several iterating steps are required to derive the averages. In other words, each sensor node should exchange the message = (yki jk ; Ykijk ) with its neighbors several times between any two subsequent time instants of the UKF. This is also the main limitation of the proposed filter since iterated exchange of messages among nodes is enviably time and energy consuming. In addition, it is difficult to know how many steps are necessary to reach consensus in practical applications. Remark 2: One of the important problems focuses on the design of the weight matrix 4 = [ ij ]N 2N that yields the fastest convergence rate for the consensus filter (17). The weight matrix 4 is usually defined by the user subject to the constraints of algebraic connectivity and graph topology. For example, the matrix 4 is stochastic and its entries ij  0. Specially, ij is strictly positive only if node i and node j are neighbors. Moreover, it has been shown that the convergence rate depends on the second largest (in magnitude) eigenvalue of 4 in the case of fixed network topology [20]. Similar results have been obtained for random network topology [21] and many accelerating algorithms have been proposed in [22]–[24]. Two well-known schemes for choosing 4 are the Maximum-degree weight and the Metropolis weight [25], where the latter is used in the simulations in this note. Remark 3: An advantage of the distributed estimation is that the property of observability of sensor nodes can be enhanced. Specifically, if the graph is strongly connected, the collective observability (i.e., when the measurements from all sensor nodes are augmented in a centralized form) should be achieved by exchanging information with neighbors even though the measurements generated by any sensor node are not sufficient to guarantee observability of the process state (i.e., local observability). The link between the observability and the network topology has been studied in [10], [26], [27].

can be approximated by a Gaussian mixture

p(xk jZk1 ; . . . ; ZkN ) =

r=1

p(xk jrk = r; Zk1 ; . . . ; ZkN )

2p(rk = rjZk1 ; . . . ; ZkN )

(20)

where Zki = fz1i ; . . . ; zki g denotes the cumulative set of measurements received from sensor i. In reality, the density p(xk jrk = r; Zk1 ; . . . ; ZkN ) is a Gaussian sum containing M k terms. Since the number of the mixture components grows exponentially with time k , a practical implementation has to limit the degree of branching in some way. The IMM estimator has been shown to be one of the most effective approximation approaches for single-sensor systems, which is based on approximating the Gaussian sum by a single Gaussian at each stage. However, in a not fully connected sensor network, each sensor node only communicates messages with its neighbors and thereby the mode-conditioned posterior density p(xk jrk = r; Zk1 ; . . . ; ZkN ) and the mode probability p(rk = rjZk1 ; . . . ; ZkN ) cannot be obtained. To this end, the consensus filters are used to generate mode-conditioned estimates and mode probabilities. Under the Gaussian approximation assumption, the first two moments of xk corresponding to the mode-conditioned posterior density p(xk jrk = r; Zk1 ; . . . ; ZkN ) can be derived by using the above distributed UKF. However, for the mode probability p(rk = rjZk1 ; . . . ; ZkN ), we have

p(rk = rjZk101 ; . . . ; ZkN01 ) p(zk1 ; . . . ; zkN jZk101 ; . . . ; ZkN01 ) 2 p(zk1; . . . ; zkN jrk = r; Zk101 ; . . . ; ZkN01 ) N p(rk=rjZk101 ; . . . ; ZkN01 ) p(zki jrk=r; Zk101 ; . . . ; ZkN01 ) i=1 = : (21) p(zk1 ; . . . ; zkN jZk101 ; . . . ; ZkN01 )

p(rk = rjZk1 ; . . . ; ZkN ) =

By defining 3ki;rjk = p(zki jrk = r; Zk101 ; . . . ; ZkN01 ), it can be obi;r served that N i=1 3kjk cannot be obtained directly by applying consensus filters since this term has a product form. To overcome this difficulty, we consider the natural logarithm of the product to transform it into a sum one N i=1

3ki;rjk = exp N

1 N1

N i=1

ki;r

(22)

i;r where ki;r ln 3ki;rjk and the average (1=N ) N i=1 k can be obi;r tained by treating k as an input state of the consensus filter (17). As the IMM approach has been well explained in [11], we will only briefly outline the basic steps in one cycle of the proposed distributed filter. The block diagram of this algorithm with two models is presented in Fig. 2, and the covariance matrices are not shown for simplicity. Note that the consensus iterating algorithm is performed between any two ^i;m subsequent time instants. Assume that the state estimate x k01jk01 , the i;m i;m covariance Pk01jk01 and the mode probability k01jk01 for node i and mode m have been obtained at time step k 0 1, this filter proceeds as follows. Step 1: Interacting of state estimates Calculate the mixing probability for sensor node i and mode r jm i;r k01jk01 =

B. Distributed Multiple Model UKF for JMNLSs By using the total probability theorem, the posterior probability density function in the case of JMNLSs with multiple sensor measurements

M

^i;r where  k01jk01 =

mr i;m k01jk01 ^i;r k01jk01

(23)

M  i;k0m1jk01 is a normalizing constant. m  =1 mr

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Step 3: Consensus update for weighted measurement and covariance



! + T ;

r = 1; 2; . . . ; M

c

yki;rjk ( +1) = yki;rjk ( )+

N

i;r ij ( )(ykj;r jk ( ) 0 ykjk ( ))

(34)

i;r ij ( )(Ykj;r jk ( ) 0 Ykjk ( ))

(35)

j =1

i;r Yki;r jk ( +1) = Ykjk ( )+

N j =1

i;r i;r i where yki;r ]01 Pxz;k [Rr;k ]01 (zki 0 z^ki;r jk (0)i;r = [Pi;r kjk01 jk01 ) and i;r 0 1 i 0 1 i;r T 01 . Ykjk (0) = [Pkjk01 ] Pxz;k [Rr;k ] [Pxz;k ] [Pki;r ] jk01 Step 4: Mode-conditioned update Calculate the updated mean and covariance i;r x^i;r = x^i;r + N Pki;r kjk jk y^kjk kjk01

i;r 01 i;r Pki;r jk = [Pkjk01 ] + N Y^kjk

Fig. 2. Block diagram of the proposed filter with two models.

With the mixing probabilities obtained in (23), estimates of all filters at time k 0 1 are mixed to obtain the initial estimates for each filter at time k

x^0ki;r 01jk01 = i;r Pk00 1jk01 =

M m=1 M

(24)

jm i;m 0i;r i;r Pki;m 01jk01 +[^xk01jk01 0 x^k01jk01] k01jk01

m=1

2 [^x 01j 01 0 x^0 01j 01] i;m k k

i;r k k

T

(25)

! + T ;

ki;r ( + 1) = ki;r ( )+

i;r;s k 1 k



0j

0i;r 01 = x^k01jk01 +

i;r (n + )Pk00 1jk01

; s = 1; . . . ; n s

(27) +n 0i;r ki;r;s 01jk01 = x^k01jk01 0

i;r (n + )Pk00 1jk01

; s = 1; . . . ; n:

x^

j 01 =

Pki;r jk01 =

s=0

2n

s=0

Ws fr (

0 j 01 )

(29)

i;r Ws [fr (ki;r;s 01jk01 ) 0 x^kjk01 ]

2 [f ( 01j 01 ) 0 x^ j 01 ] i;r;s k k

r

i;r k k

T

+ Qr;k01 :

(30)

x^ikjk = Pkijk =

M

i;r Pxz;k =

2n s=0

2n

s=0

Ws hri (fr (ki;r;s 01jk01 ))

(31)

i;r;s Ws [fr (ki;r;s 01jk01 ) 0 x^kjk01 ]

2 [h (f ( 01j 01)) 0 z^ j 01 ] i r

i;r = Pzz;k

2n

s=0

r

i;r;s k k

i;r k k

T

(32)

i;r Ws [hir (fr (ki;r;s 01jk01 )) 0 z^kjk01 ]

2 [h (f ( 0 j 01)) 0 z^ j 01 ] i r

r

i;r;s k 1 k

i;r k k

T

:

M

^ i;m ^i;m 3 k01jk01 k

(39)

r =1 M r =1

(40)

i;r i i;r i T i;r [Pki;r kjk jk + (^xkjk 0 x^kjk )(^xkjk 0 x^kjk ) ]:

(41)

Remark 4: Note that the proposed distributed filtering algorithm reverts to an unscented version of the well-known IMM estimator when there is only one sensor node. In particular, every node should broadi;r i;r cast  = (yki;r jk ; Ykjk ; k ) to its neighbors at each time step. IV. SIMULATION RESULTS To verify the performance of the proposed filter, we consider a twodimensional (2D) multi-static radar tracking system involving multiple UAVs tracking a maneuvering target as shown in Fig. 1[16]. The target state is denoted by xk = [px;k ; vx;k ; py;k ; vy;k ]T , where (px;k ; py;k ) and (vx;k ; vy;k ) represent the target position and velocity components in the (x; y ) coordinate system, respectively. The target dynamics is described by the coordinated turn model

xk = (33)

^ i;r ^i;r 3 k01jk01 k

i;r x^i;r kjk kjk

Calculate measurement sigma points and the cross-correlation covariance matrices

z^ki;rjk01 =

(38)

i;r k

s

i;r;s k 1 k

ij ( )(kj;r ( ) 0 ki;r ( ))

^ = expfN (Tc )g with  i;r (Tc ) being the output of the where 3 k consensus filter (38). Step 7: Combination of mode-conditioned estimates Given all the mode-conditioned fused estimates, the final state estimate is obtained as a weighted sum of individual fused estimates

Propagate sigma points to get the predicted mean and covariance 2n

j =1

m=1 i;r k

(28)

i;r k k

N

i;r i where ki;r (0) = ln 3ki;r and 3ki;r = N (zki ; z^ki;r jk01 ; Pzz;k + Rr;k ). Step 6: Mode probability update

i;r = kjk (26)

(37)

r = 1; 2; . . . ; M

c

Step 2: Mode-conditioned prediction Generate 2n + 1 sigma points as follows: 0 0i;r ki;r; 01jk01 = x^k01jk01

(36)

i;r i;r i;r where y^ki;r jk = ykjk (Tc ) and Y^kjk = Ykjk (Tc ) are the outputs of (34) and (35), respectively. Step 5: Consensus update for mode likelihood function



jm x^i;m i;r k01jk01 k01jk01

01

1 0 0 0

sin(!T ) !

cos(!T )

10cos(!T ) !

sin(!T )

0 0 1 0

0 10cos( ) 0 sin(!T ) !T

!

sin(!T ) !

cos(!T )

xk01 + wk01 (! ) (42)

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TABLE I NUMBER OF ITERATING STEPS TO REACH CONSENSUS

where ! denotes the turn rate and T = 1 is the sampling time period. wk01 (!) is zero-mean white Gaussian noise with covariance matrix

Q(!) = !2 I2

T

T

T

T

3 2

2

(43)

where I2 is the identity matrix and denotes the Kronecker product. To cover unknown target motions, three models corresponding to different turn rates are used. Model 1 is a coordinated turn model with turn rate 0 =s and 0 = 2. Model 2 is a coordinated turn model with clockwise turn rate 10 =s and 10 = 3. Model 3 is a coordinated turn model with counterclockwise turn rate 10 =s and 10 = 3. The switching between three models is governed by a first order homogeneous Markov chain with known transition probabilities ii = 0:8 (i = 1, 2, 3) and ij = 0:1 (i 6= j ). i ; pi ) denote the position of the ith UAV at time k , and asLet (px;k y;k sume that every UAV is moving in uniform circular motion with radius of Rc = 600 and speed V . The heading angle 'ik is measured counterclockwise from the x-axis. It should be mentioned that the speed V and the heading angle 'ik depend on the number of UAVs so that their trajectories form a circle over all time steps. Then the position of the ith UAV is approximated by i i i px;k = px;k 01 + V T cos 'k i i i py;k = py;k01 + V T sin 'k :

(44) (45)

The time-delay and the Doppler measurement equations can be written as (46)–(47) [16], shown at the bottom of the page, where c is the speed of light, and  is the wavelength of the signal. They are taken to be 3 2108 m=s and 0.3 m, respectively. The covariance matrix of the measurement noise is chosen as Rki = diagf1005 ; 125g for all UAVs. In the simulations, the sampling length is 60. We consider a family of undirected ring graphs in which the ith UAV can communicate with the (i 0 j )th and the (i + j )th UAV where j 2 f1; 2; . . . ; nb g. In other words, the number of neighbors for each UAV is Nb = 2nb . The centralized filter serves as the baseline algorithm and the proposed filter is compared with it. To analyze the tradeoffs between the number of nodes N , the number of neighbors Nb and the number of consensus iterating steps Tc , the required number of iterating steps to reach consensus is shown in Table I

ki = 1 c

2 + p2 + px;k y;k

y;k

Fig. 4. Mode estimation of Model 1.

for different combinations of N and Nb . It can be observed that Tc increases fast as N increases for fixed Nb and Tc deceases fast as Nb increases for fixed N . This is expected since more information can be obtained by each node if the number of neighbors increases and thus less consensus iterating steps are needed. In addition, it seems that Tc is proportional to the ratio of Nb =N . For instance, Tc is around 15 when the ratio Nb =N 2 [1=2; 2=3]. It should be pointed out that the required number Tc might still be too large for practical applications. Thus,

i )2 + (p i (px;k 0 px;k y;k 0 py;k )2

fki = 1 vx;k px;k + vy;k py;k  p2 + p2 x;k

Fig. 3. Performance comparison w.r.t RMSE in position.

i i i i + (vx;k 0 V cos 'k )(px;k 0 pix;k )2+ (vy;k 0 V isin '2 k )(py;k 0 py;k ) (px;k 0 px;k ) + (py;k 0 py;k )

(46)

(47)

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TABLE II PERFORMANCE COMPARISON W.R.T ARMSE

proposing accelerating consensus strategies has been another topic for future research [22]–[24]. To evaluate the performance of the algorithms, the averaged root mean square error (ARMSE) in position is defined as the average of RMSE in position over all nodes and all time steps. This definition is similar to the network mean square deviation (MSD) in [8]. Simulation results are derived from 50 Monte Carlo runs. From Table II, the following features can be deduced. First, the ARMSE in the case the sensors do not communicate with each other (i.e., b = 0) decreases as increases, and it is greater than the other values with fixed . Secondly, as shown in the centralized estimation, more accurate tracking performance can be achieved by increasing the number of nodes . This is consistent with conventional results for multi-sensor fusion. Thirdly, the performance of the distributed filter is comparable to that of the centralized filter if the consensus has been reached and the performance can not be improved by more consensus iterating steps once the consensus is reached. Fourthly, the performance can not be improved by adding more nodes if b and c are fixed. Therefore, by considering the requirement of computation and communication, a modest conclusion can be drawn that it is better to increase the number of neighbors than to increase the number of nodes for guaranteeing tracking performance. Specially, the performance of the proposed filter and the centralized = 10, b = 4 and c = 30, which filter is shown in Fig. 3 with indicates that they provide almost identical estimates. Moreover, these results also suggest that the consensus has been reached for local filters. The comparison of the mode estimation of Model 1 with the truth is presented in Fig. 4. It can be seen that the true mode can be identified by each node. Note that for the sake of clarity, we have omitted the performance of UAV4 -UAV10 in Figs. 3 and 4.

N

N

N

N

N

T

N

N

T

V. CONCLUSION In this note, a distributed multiple model UKF has been developed for discrete-time JMNLSs. This filter is derived by utilizing the IMM approach, in which the mode-conditioned estimates and mode probabilities are computed in a distributed manner using consensus filters. The resulting algorithm requires no global models in addressing the issue of multiple model fusion and each sensor node only communicates with its neighbors. Thus, it is scalable and robust. Simulation results show that the performance of the proposed filter is in close agreement with that of the centralized filter where all sensor nodes can communicate with each other.

ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments.

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-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils Peter Benner, Vasile Sima, Senior Member, IEEE, and Matthias Voigt

Abstract—In this technical note, we discuss an algorithm for the compu-norm of transfer functions related to descriptor systems. tation of the We show how one can achieve this goal by computing the eigenvalues of certain skew-Hamiltonian/Hamiltonian matrix pencils and analyze arising problems. We also formulate and prove a theoretical result which serves as a basis for testing a transfer function matrix for properness. Finally, we illustrate our results using a descriptor system related to mechanical engineering. control, numerical staIndex Terms—Continuous time systems, bility, singular systems, skew-Hamiltonian/Hamiltonian matrix pencils, transfer function matrices.

I. INTRODUCTION In many applications from industry and technology, computer simulations are performed using models which can be formulated by systems of differential equations. Often the equations underlie additional algebraic constraints which prevent the system from attaining every possible state. In this context we speak of descriptor systems (or sinManuscript received September 29, 2010, revised March 08, 2011; accepted July 05, 2011. Date of publication July 14, 2011; date of current version December 29, 2011. This work was supported by the Deutsche Forschungsgemeinschaft under grant BE-2174/6-1. Recommended by Associate Editor F. Dabbene. P. Benner and M. Voigt are with the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg 39106, Germany (e-mail: [email protected]; [email protected]). V. Sima is with the National Institute for Research and Development in Informatics, Advanced Research Department, Bucharest 011455, Romania (e-mail: [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2161833

gular systems). These systems naturally arise in a large variety of applications such as electrical circuit simulation, multi-body dynamics with constraints or the semidiscretization of certain partial differential equations (see [1] and references therein). Very important characteristic values of such systems are the L -norms of the corresponding transfer functions. These norms have found important applications in robust control or model order reduction [1]–[3]. Consider a continuous-time linear time-invariant descriptor system

1

Ex t with E , A 2

Ax t Bu t ; y t Cx t Du t (1) n2n , B 2 n2m , C 2 p2n , D 2 p2m , descriptor vector x t 2 n , control vector u t 2 m , and output vector y t 2 p . Here, E usually is a singular matrix. By taking the Laplace _( ) =

( ) +

( )

( ) =

( )

( )+

( )

( )

( )

transform [4] of both equations in (1) and inserting the first equation into the other one we obtain the matrix-valued transfer function of the system

Gs

( ) :=

C sE 0 A 01 B D (

)

(2)

+

12

12

which directly maps inputs to outputs in the frequency domain. For convenience, we assume that G 2 RLp m , where RLp m denotes the Banach space of all rational p 2 m matrix-valued functions that are bounded on the imaginary axis. The natural norm for this space is the L -norm which is defined by

1

kGkL

!2

:= sup

max G i! (

(

))

where max denotes the maximum singular value [2]. Note that for any transfer function G 2 RLp m , the matrix pencil E 0 A is regular (i.e., det(E 0 A) is not identical to the zero polynomial) and does not have any finite eigenvalues on the imaginary axis. However, even if E 0 A is regular and has no finite purely imaginary eigenvalues, the transfer function could still be unbounded at infinity and hence is not an element of the space RLp m . This motivates the following definition. kG(i!)k < 1 and We call a transfer function G proper if lim! strictly proper if lim! kG(i!)k = 0 for any induced matrix norm k 1 k. Otherwise we call it improper [5]. As an agreement, we assign the norm value 1 to transfer functions G 62 RLp m . In the sequel we do not assume that the system is stable. However, if it is stable, the L -norm is equivalent to the H -norm [2]. The implementation based on the conceptual algorithms proposed in this technical note covers the case of systems with unbounded transfer functions. The check for properness is done optionally. If the test is skipped, an error indicator is set for an improper system. A regular matrix pencil E 0 A can be reduced to Weierstraß canonical form [5]

12

12 !1

!1

12

1

1

E W In

J N T; A W In T where W and T are nonsingular, Im is an identity matrix of order m, J and N are in Jordan canonical form and N is nilpotent with index of nilpotency  . The number  is also called the algebraic index of the descriptor system (1) and nf and n1 are the dimensions of the deflating subspaces of E 0 A corresponding to the finite and infinite eigenvalues, respectively. By using the transformation matrices W and T we can also write B and C as 1 01 B W 01 B B2 ; C C1 C2 T : =

0

=

0

=

0

0

= [

]

In this way the system (1) is decoupled into its slow subsystem

x1 t _

0018-9286/$26.00 © 2011 IEEE

( ) =

Jx1 t

( )+

B1 u t ; y1 t ( )

( ) =

C1 x1 t

( )