J. Cent. South Univ. (2013) 20: 2186−2194 DOI: 10.1007/s11771-013-1723-4

Optimal sensor scheduling for hybrid estimation

LIU Jian-liang(刘建良)1, SUN Yao(孙尧)1, YANG Jian(杨建)1, LIU Wei-yi(刘为夷)2, CHEN Wei-min(陈卫民)1

1. School of Information Science and Engineering, Central South University, Changsha 410083, China;

2. Qualcomm Inc, Santa Clara 95051, USA

© Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: A sensor scheduling problem was considered for a class of hybrid systems named as the stochastic linear hybrid system (SLHS). An algorithm was proposed to select one (or a group of) sensor at each time from a set of sensors. Then, a hybrid estimation algorithm was designed to compute the estimates of the continuous and discrete states of the SLHS based on the observations from the selected sensors. As the sensor scheduling algorithm is designed such that the Bayesian decision risk is minimized, the true discrete state can be better identified. Moreover, the continuous state estimation performance of the proposed algorithm is better than that of hybrid estimation algorithms using only predetermined sensors. Finally, the algorithms are validated through an illustrative target tracking example. Key words: sensor scheduling; hybrid systems; Bayesian decision risk; target tracking

1 Introduction

The problem of sensor scheduling involves utilizing multiple sensing agents to estimate the true state of a system. This problem arises in many applications: many modern radars have several working modes and the controller can decide which mode to be used to give the best observation data [1]. Some active sonars may interfere with each other when they are operating at the same time, which naturally prevents them from working simultaneously [2]. It may be costly or unnecessary to employ all the sensing agents all the time to provide the target information, e.g., sensor networks [3]. By properly switching between different sensors and merging or exchanging the information between them, sensor scheduling often gives more target information and better estimation accuracy but increases the complexity of the whole system. Thus, to design effective sensor scheduling algorithms is important in many applications.

Generally speaking, sensor scheduling can be regarded as an optimal control problem which involves deriving optimal control logic for the sensors such that some cost (i.e. the estimation error) is minimized. The seminal work in this area can be found in Refs. [4−6]. It has been proved that this problem can be formulated as a two-point boundary value problem [7], or can be solved by the greedy algorithm [8]. Recently, SINOPOLI et al

presented interesting results (see Refs. [9−12] and the references therein). Also, HE and CHONG [13] have successfully applied the Monte Carlo method to the sensor scheduling problem. Other relative research can be found in Refs. [14−16].

A system with switching sensors can be modeled as a hybrid system, a class of system with interacting continuous dynamics and discrete dynamics. Also, switching between different sensors can be regarded as a transition between different discrete states of a hybrid system. In Refs. [17−18], the sensor scheduling problem is solved using the hybrid systems’ approaches, but the nominal system itself is a linear-time-invariant (LTI) system. This work considers the case in which the nominal system is a hybrid system with multiple switching modes, each observed by multiple sensing agents (Fig. 1). Past research has shown that the state estimation itself is challenging for hybrid systems with synchronized sensors. If the discrete state transition history is unknown, the evolution of a hybrid system involves the exponentially increasing number of hypotheses over time [19]. This makes “optimal estimation” of the hybrid system computationally prohibitive. In this work, the optimal sensor scheduling algorithm can estimate the discrete state and the continuous state of the SLHS. The proposed algorithm is divided into two parts: first, the optimal switching logic (switching history) of the sensors is determined through

Foundation item: Project(2012AA051603) supported by the National High Technology Research and Development Program 863 Plan of China Received date: 2012−04−23; Accepted date: 2012−07−26 Corresponding author: YANG Jian, Associate Professor, PhD; Tel: +86−731−88876070; E-mail: jian.yang@csu.edu.cn

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Fig. 1 Configuration of stochastic hybrid system, sensor

network and hybrid estimator

a sensor scheduling algorithm; second, the hybrid estimation is carried out based on the optimal sensor switching history. Due to the complexity of the whole system involving both switching system dynamics and switching observation model, the computation of the optimal senor switching logic is intractable if it is designed such that the estimation error is minimized directly. To avoid this difficulty, the “best” switching logic is chosen in the sense that the Bayes risk in the estimation algorithm is minimized. The Bayes risk is parameterized as the common area (or volume) under the different likelihood functions of the different hypotheses. Although there is no closed-form solution, its upper bound can be easily computed. In the case of two hypotheses testing, the area is bounded by the Battacharyya Bound [20]. The Bayesian decision risk bound of multiple hypotheses can be found in Ref. [21]. The optimal sensor switching logic is designed such that the Bayesian decision risk bound is minimized at each estimation step. To demonstrate the performance of our algorithms, they are tested in anillustrative target tracking scenario in which two sensors work cooperatively to estimate the true state of a maneuvering target whose dynamics can be modeled as a stochastic linear hybrid system. Simulation results show that by properly choosing the best sensor at each time step, the proposed algorithm identifies the discrete state more clearly and gives the better continuous state estimation result than the algorithm that uses one predetermined sensor only, which is in accordance with the theoretic results. 2 Problem formulation

In this section, the stochastic linear hybrid system (SLHS) [22] is reviewed. Then, a switching sensor model, using multiple sensors for hybrid estimation, is presented. 2.1 Stochastic linear hybrid system

Considering the following stochastic linear hybrid system (SLHS) model, the discrete-time continuous

dynamics of the SLHS is given by

( ) ( ) ( )( ) ( 1) ( ) ( )q k q k q kk k k k x A x B u F w (1) where x(k)X=Rn is the state vector; R umu is the known input vector; q(k) Q=1, 2, … , nd is the discrete state at time k; Q is a finite set of all the discrete states; Aq, Bq and Fq are the system matrices with appropriate dimensions, corresponding to each discrete state q Q, and ( ) wmw k Q is the white Gaussian process noise with w(k)~N(0, W(k)).

There are two types of discrete state transitions in the SLHS:

1) Markov-jump transition model: the discrete state transition history is a realization of a homogeneous Markov chain. The finite state space of the Markov chain is the discrete state space Q. Suppose at each time k, the probability vector is given by 1( ) [ ( )k kπ π … T( )]

dn kπ , with

πi(k) denotes the probability that the system’s true

discrete state is i. Then, at the exit time step, the probability vector is

( 1) ( )k k π Γπ (2) where a constant matrix Γ is the Markov transition matrix with 1ij

j

Γ (we use ijΓ to denote the scalar

component in the i-th row and j-th column in the Markov transition matrix Γ ). Here, the discrete dynamics is decoupled from the continuous dynamics, i.e., the discrete transition is independent of the continuous dynamics. This model has been widely used due to its computational simplicity.

2) State-dependent transition model: the discrete state transition is governed by

( 1) ( ( ), ( ), )q k q k x k (3) where Rl Θ and : X Q Θ Q is the discrete- state transition function defined as

TT T( , , ) , if [ ] ( , )i j G i j x θ x θ

G(i, j) is called the guard condition. For each combination of (i, j), the guard condition G(i, j) is a subset of the space X Ω Θ with the following assumption.

Assumption 1: The set of guards ( , ) | , G i j i jQ

is a partition of the space for any given iQ :

, , , , and ( ) ( ) , i j i k i j k j kG G Q

(4) and

1

( , ) r

j

G i j i

Q (5)

In this work, we consider a specific kind of the

guard condition ( , ) | ,G i j i jQ named as the stochastic linear guard condition:

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( , ) , , 0ij ij

x xG i j x X θ Θ L b

θ θ

(6)

where

l θ Θ R and θ~N( , Σθ) is a dimensional Gaussian random vector with mean and covariance Σθ representing uncertainties in the guard condition; Lij is a v×(n+1) matrix, bij is a constant v-dimensional imensional vector, and ν is the dimension of the vector inequality. Here, a vector inequality y ≤ 0 means that each scalar element of y is non-positive.

Remark 1: The state-dependent transition history is a realization of a Markov chain, but not a homogeneous Markov chain. The transition matrix Γ is not constant any more but a function of x(k) and θ at each time and can be written as ( ( ), )kΓ x θ . From Assumption 1, it can be proved that the matrix ( ( ), )kΓ x θ is a Markov

transition matrix with ( ( ), ) 1ijj

k Γ x θ [22].

2.2 Observation model

In the observation model, several sensing agents are working cooperatively to provide the observation information. Suppose the cardinal number of the sensor set is M. The following assumption is made:

Assumption 2: At each time k, only one sensor is operating from a set of M sensors.

Assumption 2 is a common assumption in the research area of sensor scheduling. It can be extended to a more general case: if we allow several sensors working together at the same time, we can stack the observation vector from each sensor to get a bigger observation vector, and treat this as observation coming from a new fictional sensor. Thus, Assumption 2 still holds.

The observation model of the i-th sensor is given by

( ) ( ) ( )i i iz k C x k v k (7) where iM, M=1, …, M is the set of M sensors;

( ) R piz k is the measurement (output) of the i-th

sensor; Ci are the observation matrices with appropriate dimension and ( ) R pv k is the white Gaussian observation noise with v(k)~N(0, Ri(k)). We use MN to denote the set of all ordered sequence of sensor schedules up to time N. Thus, an element 0 ,N N

1 1, , N N Nk M is a N-horizontal sensor schedule.

Under a given sensor schedule σN, the measurement sequence is given by

( ) ( ) ( ) ( ),N N Nk k

z k z k C x k v k

0, 1, , 1k N

(8)

3 Bayes risk hybrid estimation

In this section, the problem of the Bayes risk is considered for hybrid estimation. Most hybrid estimation

algorithms [19] involves a Bayes decision procedure of computing the likelihood of each discrete state based on the new measurement at each time step. Generally speaking, the purpose of sensor scheduling is to minimize the Bayes risk in this procedure. The Bayes risk can be parameterized as the area (volume) under multiple hypotheses whose analytical solution is intractable. Thus, to compute the upper bound of the Bayes risk, we use the Battacharyya bound which can be easily computed. 3.1 An illustrative example

Consider a stochastic hybrid system whose continuous state consists of two elements x=[x1 x2]

T and its one-step evolution is shown in Fig. 2. Suppose the system has two discrete states Q=1, 2 and at time k, the continuous state probability distribution function (PDF) is given by p[x(k)]1. Without loss of generality, we assume the distributions are Gaussian and the system is a linear hybrid system. At time k+1, before the arrival of measurement zi , we propagate the system under the two hypotheses: H0: q(k + 1) = 1 and H1: q(k + 1) = 2

The PDFs of p[x(k+1)|H0]PH0 and p[x(k+1)| H1]PrH1 are shown in Fig. 2 using their 1−σ ellipsoid. Suppose we choose only one sensor to provide the measurement at each time, the sensor model is given by

1 1 1 1( ) [1 0] ( ) ( ) ( ) ( )z k x k v k x k v k

(9)

2 2 2 2( ) [0 1] ( ) ( ) ( ) ( )z k x k v k x k v k (10) where v1 and v2 are the observation noises. From Eqs. (9) and (10), we can compute the likelihoods of the two hypotheses H0 and H1 under the condition that Sensor 1 or 2 is currently used, which is shown in Fig. 3. Figure 3(a) shows the likelihood functions of the two hypotheses when Sensor 1 is activated, and Fig. 3(b) shows the likelihood functions when Sensor 2 is activated. In Fig. 3, the difference between the likelihoods in Fig. 3(a) is much bigger than that in Fig. 3(b), i.e., Sensor 1 can clearly identify the true discrete state under this scenario. The probability of

Fig. 2 Constinuous evolution of simple stochastic linear hybrid

system

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Fig. 3 Bayes risk of two sensors: (a) Sensor 1; (b) Sensor 2

making a wrong decision in this scenario is regarded as the Bayes risk. The shaded area is a measure of the Bayes risk of the two hypotheses scenario. Obviously, the probability of the Bayes decision error is low if the shaded area is small. 3.2 Bounding Bayes risk

The general hypotheses testing problem is shown in Fig. 4. Given a general observation Z and a general sensor schedule σ, the Bayes decision involves multiple decision regions Ri, where Hi is the most likely hypothesis:

| [ , | ] [ , | ], i i jR Z p H Z p H Z j i (11)

Thus, the Bayes risk can be computed by

, | j ii j i

P error P Z R H

, | j i ii j i

P Z R H P H

| , dj

i iRi j i

p Z H P H Z

(12)

where the probability Perror (Bayes risk) can be regarded as the objective function to be minimized for the sensor scheduling algorithm. However, Eq. (12) is computationally intractable. It is possible to bound the Bayes risk in a closed form. If there are only two

Fig. 4 Likelihood functions for general Bayes decision scenario

hypotheses H0 and H1, the Battacharyya Bound can be applied, which is given by

ZHZpHzpHPHPerrorP d]|[]|[)( 102/1

12/1

0

(13) If the likelihood functions of the both hypotheses

are Gaussian: 0 0 0[ | ] ~ ( , )p Z H N and 1[ | ] ~p Z H

1 1( , ),N integral Eq. (13) can be evaluated

analytically:

1 1

2 20 1 exp P error P H P H k (14)

where

T 11 0 0 1 1 0

0 1

0 1

1[ ] [ ] [ ]

4( ) / 21

ln2

k

u u u u

Note that in Eq. (14), the upper bound of the probability Perror can be regarded as the cost that the sensor scheduling algorithm is to minimize. If the Bayes decision involves multiple hypotheses, each being a Gaussian distribution: ,[ | ] ( )i i ip Z H N u , the Bayes decision upper bound can be computed by [21]

1 1

2 2 exp ( , )i ji j i

P error P H P H k i j

(15)

where

T 11( , ) [ ] [ ] [ ]

4

( ) / 21 ln

2

j i i j j i

i j

i j

k i j

u u u u

In summary, we have derived a relationship

between the upper bound of the Bayes risk and the sensor selection problem for hybrid estimation.

4 Sensor scheduling and hybrid estimation algorithm

In this section, we present a sensor scheduling

algorithm combined with the hybrid estimation algorithm. Our algorithm uses the “mixing” step, which is similar to the IMM algorithm [23] to keep the exponentially growing computational complexity constant. For the Markov-jump transition model, the a priori knowledge about the discrete state transition is given by the Markov transition matrix, which assumes the discrete state transition probabilities Pq(k+1)=j|q(k)=i, Zk, σk to be constant. However, in the state-dependent transition model, the discrete state transition probabilities are computed from the continuous state estimates.

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Then, using the discrete state transition probabilities, we can compute the initial conditions for a bank of Kalman filters (KF) in which each KF is matched to a discrete state of the hybrid system. Based on the initial conditions, the sensor scheduling algorithm decides which sensor should be used in the next step, so that the Bayes risk is minimized. Then, the “best” sensor is turned on; the new measurement arrives and is fed into each KF which updates the continuous state estimate and computes the likelihood of each discrete state.

Finally, the estimate of the continuous state is given by a weighted sum of the output of each KF, and the discrete state estimate is given by the discrete state with the highest probability among all discrete states. Each step of our algorithm is described as follows (Fig. 5).

Step 1: Mixing (merging) probabilities

The mixing probabilities Pq(k)=i|q(k+1)=i, Zk, σk

for all i, jQ are defined as

( ) : ( ) | ( 1) , ,k kji k P q k i q k j Z (16)

By the Bayes’ theorem,

( ) | ( 1) , , k kP q k i q k j Z

1

( 1) | ( ) , , [ ( ) | , ]k k k k

j

P q k j q k i Z p q k ZC

(17) where Cj is a normalizing constant. To evaluate Eqs. (16) and (17), we use the following approach to compute the discrete state transition probability Pq(k+1)= j|q(k)=i, Zk, σk:

Fig. 5 Proposed sensor scheduling and hybrid estimation algorithm

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1) Markov-jump transition: The Markov transition matrix provides the a priori

knowledge directly. The discrete state transition probability Pq(k+1)=j|q(k)=i, Zk, σk in Eq. (17) can be written as: Pq(k+1)= j|q(k)=i, Zk, σk=Γij=const (18)

2) State-dependent transition: We recall that ~ ( , )l N Θ R has a

multivariate Gaussian distribution (if Θ≠0). With the linear guard condition given in Eq. (6), we compute the discrete state transition probability Pq(k+1)= j|q(k)=i, Zk, σk=Γij=const in Eq. (17) as [22]

,,)()1( kkZikqjkqP

T

0

0)(,

)(ˆij

iijijijv

kPb

kxΦ LLL

(19)

where ),( yv yΦ is the v-dimensional Gaussian

cumulative density function (CDF) with mean y and

covariance Σy:

)0(:),( yPy yv

0

0

0

21 ddd) , ;( vv yyyyyN

Step 2: Initial conditions for each KF At each time k, we approximate the initial condition

of each KF by a single Gaussian distribution. The initial conditions (mean )(ˆ 0 kx j and covariance Pj0(k)) for the j-th KF are given by

dn

ijij kkk

10 )(ˆ)()(ˆ xμx (20)

)(ˆ)][(ˆ)(ˆ[)()()( 01

0 kkkkkk ijii

n

ijij

d

xxxPμP

)](ˆ T 0 kjx (21)

Step 3: Sensor scheduling For mode j, compute the prior distribution

(likelihood function) of the observation zi(k+1) (recall that the subscript i means the i-th sensor is used for observation):

))1(,)1((],,)1()1([ kkNZjkqkzp ijijkk

i Sr

(22) where

)()1()()1(

)1()(ˆ)1(

TTT0

0

kkkk

kkk

iiiijjjiij

jijjiij

RCQCCAPACS

uBCxACr

Based on the likelihood function

( 1) | ( 1) , ,k kip z k q k j Z ,

compute the Battacharyya Bound of the Bayes risk of the i-th sensor using Eqs. (14) or (15). The sensor schedule σk is augmented to σk+1 such that the upper bound in Eq. (14) or Eq. (15) is minimized at time k+1.

Step 4: Mode-matched filtering Suppose at time k + 1, the l-th sensor in the sensor

set M is chosen in Step 3. After the arrival of the new measurement zl(k), each KF computes the posterior mean and covariance )1(ˆ kjx , )1( kjP conditioned on

( )q k j for .Qj Step 5: Discrete-state PDF update For each KF, the likelihood function is

))1(,0;)1((

,,)1()1(:)1( 1

kkN

ZjkqkzpkΛ

ijijp

kkj

Sr

(23)

By Bayes’ theorem, the discrete state probability

1 1( 1| 1) : ( 1) | ,k kj k k P q k j Z

is given by

)1(|)1(1

)1|1( kqkzPkk lj

j

,|)1(,, 11 kkkk ZjkqPZj (24) where δj is a normalizing constant. Substituting Eq. (23) into Eq. (24) and using the total probability theorem on the term Pq(k+1)=j|Zkσk+1 in Eq. (24), we get

1( 1| 1) ( 1)j j

j

k k k

1

1

( 1) | ( ) ; , dn

k k

i

P q k j q k i Z

1 ( ) | , k kP q k i Z (25)

Step 6: Output By the total probability theorem, the continuous

state PDF at time k + 1 is given by

1 11 | ,k kp k Z x

1 1

1

1 | 1 , ,dn

k k

j

p x k q k j Z

],|)1([ 11 kkZjkqp (26)

We approximate the sum of the r terms in Eq. (26) via moment matching by a single Gaussian PDF [24]:

))1(,)1(ˆ;(],|)1([ 11 kPkNZkp nkk xxx

where

1

1

T

ˆ ˆ( 1) ( 1 1) ( 1 1)

( 1) ( 1 1) ( 1 1)

ˆ ˆ ˆ ˆ [ ( 1 1) ( )][ ( 1 1) ( )]

d

d

n

j jj

n

j jj

j j

k k k k k

P k P k k k k

k k k k k k

x x

x x x x

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The discrete state PDF at time k + 1 is given by

)11(,)1( 11 kkZjkqP jkk

and its estimate is:

11,)1(maxarg)1(ˆ kk

jZjkqPkq

5 Simulations

In this section, the performance of our algorithm through a target tracking scenario is validated. 5.1 Target dynamics

For the purpose of illustration, we consider the dynamics of a target which has three discrete states (modes): left turn (LT), right turn (RT) and constant velocity (CV) as shown in Fig. 6. In the LT mode and the RT mode, the target performs a coordinated turn with a constant turning rate while the target keeps its velocity constant in the CV mode. We assume that the transition between different modes to be governed by a time-homogeneous Markov Chain whose evolution is given by Eq. (2). Let ,])()()([)( T

321 kkkk π where π1(k), π2(k) and π3(k) are the probabilities that the true discrete state is LT, RT or CV, respectively. The Markov transition matrix is parameterized as

0.6 0.2 0.2

0.2 0.6 0.2

0.2 0.2 0.6

Γ (27)

Fig. 6 Discrete state transition model of target

The continuous state of the target is represented by

the state vector: 4T[ ] X x R in the

ξ−η frame. The continuous dynamics is governed by stochastic difference equations, each corresponding to one discrete state. The dynamics corresponding to the

CV mode is given by Ref. [22]:

)(

)(

)(

)(

1000

100

0010

001

)1(

)1(

)1(

)1(

S

S

k

k

k

k

T

T

k

k

k

k

)(

)(

2

0

0

0

0

2

S

2SS

2S

kw

kw

T

TT

T

CV

CV

(28)

where wξCV and wηCV are independent white Gaussian noise with 2 2[ ( )] 1 ,CVE w k 2 2[ ( )] 1CVE w k and

0])()([ 2 kwkwE CVCV ; Ts=1 s is the sampling time.

The continuous dynamics corresponding to the LT and

the RT mode is given by [22]

S S

S S

S S

S S

sin( ) 1 cos( )1 0

( 1)

0 cos( ) 0 sin( )( 1)

1 cos( ) sin( )( 1)0 1

( 1)0 sin( ) 0 cos( )

T Tk

T Tk

T Tk

kT T

2S

/ RT2S S

/ RT

S

0( )

02 ( )( )

( )( )0 2

( )0

L

L

Tk

w kkT T w kk

kT

(29)

where w=10 (°)/s for the LT mode and w=−10 (°)/s for

the RT mode; wξL/RT and wηL/RT are independent white

Gaussian noise with ,1])([ 22RT/ kwE L

22RT/ 1])([ kwE L and )([ RT/ kwE L 0)](RT/ kw L .

We assume that there are two sensing agents providing the observation information. At each time k, only one sensor is turned on. The observation model for Sensor 1 is given by

)()()( 111 kkCkz vx

)(

)()(

00.110

0.1001

12

11

kv

kvkx (30)

and the observation model for Sensor 2 is given by

)()()( 222 kkCkz vx

)(

)()(

010.10

1000.1

22

12

kv

kvkx (31)

In Eqs. (30) and (31), v11, v12, v21 and v22 are

mutually independent white Gaussian noise with 2 2 2 2 211 12 21 22( ) ( ) ( ) ( ) 1 .E v k E v k E v k E v k

The target motion is simulated for 110 s. Figure 7

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shows the actual target trajectory and compares it with the position estimation results computed by different observation setups: two sensors working cooperatively, using Sensor 1 only, and using Sensor 2 only. From Fig. 7, we can see that the proposed hybrid estimation algorithm combined with the sensor scheduling algorithm gives the best position estimates. If a predetermined sensor is applied, the Bayesian decision risk for the sensor is high, which leads to big position estimation errors. However, if the two sensors working cooperatively, the Bayesian decision risk is always reduced to a low level.

Figure 8 shows the discrete state probabilities and Figure 9 shows the discrete state estimates given by the three algorithms. From Figs. 8 and 9, we can see that the true discrete state can be clearly identified if the two sensors are scheduled properly.

Figure 10 compares the RMS position estimation errors obtained via a 100 run Monte Carlo simulation. In Fig. 10, the RMS errors of the sensor scheduling algorithm stays below the error of the hybrid estimation algorithms that utilize one predetermined sensor. The simulation shows that the sensor scheduling algorithm

Fig. 7 Target trajectory and position estimation results

Fig. 8 Discrete state probabilities

Fig. 9 Discrete state estimation

Fig. 10 Result of Monte Carlo simulation: Root-Mean-

Square(RMS) error of position estimates

selects the “best” sensor such that the estimation error is minimized at each step. Table 1 compares the discrete state estimation errors and summarizes the continuous state estimation errors given by the three algorithms. From the simulation results, we can see that the hybrid estimation algorithm with scheduled multiple sensors is the best among the three algorithms.

Table 1 Performance comparison of statistics of 100 simulation

runs

RMS position error/m

Algorithm Peak

Overall average

Average No. of discrete state

estimation error

Sensor scheduling 4.335 4 3.523 3 17

Sensor 1 only 7.524 4 4.432 7 35

Sensor 2 only 5.838 2 4.235 9 33

6 Conclusions

In this work, the problem of optimal sensor scheduling has been considered for hybrid state

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estimation. The sensor scheduling problem involves using several sensing agents cooperatively to provide the observation information and estimate the continuous state and the discrete state of a hybrid system. However, sensor scheduling for hybrid systems is challenging in that the hybrid state estimation involves an exponentially growing hypothesis tree. To reduce the computational complexity, the Bayes risk is used for the hybrid estimation with multiple sensors. The optimal sensor scheduling is designed such that the Bayes risk (or its upper bound) is minimized. Thus, combining with the proposed sensor scheduling algorithm, our hybrid estimation algorithm computes the accurate hybrid state estimate. The performance of the proposed algorithm has been validated through a target tracking scenario. Simulation results show that our algorithm can schedule the optimal sensor history such that its performance is better than the hybrid estimation algorithms that use the predetermined sensor only. References [1] KRISHNAMURTHY V, DJONIN D V. Optimal threshold policies

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(Edited by HE Yun-bin)