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Particle filtering for networked nonlinear systems subject to random one-step sensor delay and missing measurements
Communicated by Dr. Ma Lifeng Ma
Accepted Manuscript
Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements Long Xu, Kemao Ma, Wenshuo Li, Yurong Liu, Fuad E. Alsaadi PII: DOI: Reference:
S0925-2312(17)31710-1 10.1016/j.neucom.2017.10.059 NEUCOM 19039
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
16 August 2017 12 October 2017 30 October 2017
Please cite this article as: Long Xu, Kemao Ma, Wenshuo Li, Yurong Liu, Fuad E. Alsaadi, Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.10.059
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ACCEPTED MANUSCRIPT
Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements Long Xu a , Kemao Ma a,∗ , Wenshuo Li b , Yurong Liu c,d , Fuad E. Alsaadi d Department of Control Science and Engineering, Control and Simulation Center, Harbin Institute of Technology, Harbin 150080, China b
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China c
Department of Mathematics, Yangzhou University, Yangzhou 225002, China
Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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Abstract
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In this paper, the particle filtering problem is investigated for a class of networked nonlinear systems with random one-step sensor delay and missing measurements. The phenomena of missing measurements and random one-step sensor delay are modeled by two random variables, both obeying the Bernoulli distribution. Here, we derive an explicit expression for the likelihood function when the possible occurrence of one-step sensor delay and measurement loss is taken into consideration. Based on this likelihood function, we propose a novel particle filtering algorithm to treat the nonlinear estimation problem in the simultaneous presence of random sensor delay and measurement loss. Finally, a simulation example is given to illustrate the feasibility and advantages of the proposed filtering scheme compared with traditional particle filtering algorithm.
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Introduction
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Key words: Particle filter; networked systems; random one-step sensor delay; missing measurements; bayesian framework.
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In recent years, the nonlinear filtering problems were extensively considered due to wide applications in many fields such as target tracking, location and parameter estimation [1–6]. It is well known that Kalman first proposed the classical Kalman filter in [7]. Up to now, the Kalman filter is still widely used as a standard framework to solve the linear estimation problem. However, in a practical environment, most of the systems have nonlinear characteristics [8, 9], hence the study of nonlinear filtering problems is of important significance. A well-known method, the Monte Carlo method, was proposed in [10] and used to solve the nonlinear filtering problems. In addition, the sequential importance sam-
Preprint submitted to Automatica
pling algorithm was utilized to estimate the nonlinear system states, where the proposed algorithm takes a set of samples extracted from the proposed distribution to approximate the distribution of the target and each sample has a corresponding weight. Due to considering the problem of sample weight degradation in the sequential importance sampling algorithm, the application of sequential importance sampling algorithm in solving nonlinear filtering problem was very limited at that time. In order to avoid this case, the re-sampling technique was introduced into the Monte Carlo importance sampling process and a new filter was proposed in [11], which can effectively solve the problem of the sample weight degradation. By summarizing the sequential importance sam-
3 November 2017
ACCEPTED MANUSCRIPT pling algorithm, the concept of Sequential Monte Carlo method, namely particle filter, was proposed in [12]. The basic idea of this method is to approximate the posterior probability distribution of the system by employing a set of samples (or particles) in order to estimate the state of the nonlinear system. Subsequently, the particle filtering algorithm has been widely applied in many practical problems.
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Due to the wide application of particle filtering approach, the scholars paid close attention to the developments of particle filtering technique in many different research fields, such as modern signal processing, communication, artificial intelligence, autonomous mobile robot and target tracking. Accordingly, a large number of application results were published concerning on particle filtering algorithm. To mention a few, for the situation of randomly missing measurements in engineering practice, a filtering method was proposed in [13] for nonlinear system based on sequential importance sampling and was applied to online state estimation for nonlinear non-Gauss systems. In [14], a new hybrid filtering algorithm was proposed for a class of nonlinear systems with linear state equation and nonlinear measurement equation based on the approaches of Kalman filter and particle filter. Based on Bayesian filtering framework, a new particle filter was designed in [15] for nonlinear systems with random one-step delayed measurements, where the probability of the time-delay is unknown and is estimated by maximum likelihood criterion. When taking the multiple-step time-delay into account, the nonlinear filtering problem was discussed in [16] for nonlinear systems with multiple-step randomly delayed measurements by applying the idea as in [15]. When the data are measured by multiple sensors, the problem of distributed particle filtering was investigated in [17] for nonlinear tracking over wireless sensor networks. Moreover, a distributed particle filter was constructed in [18] for bearings-only tracking of a target moving on the surface of a sphere and a proof of uniform convergence of distributed particle filtering algorithm was given in [19].
missing measurements have attracted the attention of many scholars and a large number of results were reported [26–30]. Recently, in [31], a linear optimal filtering problem was investigated for a class of stochastic timevarying systems subject to state delay and randomly multiple sensor delays based on the innovative analysis approach and recursive projection formula. By using the same method as in [31], the optimal filter was designed in [32] for a networked stochastic state delay systems with missing measurements. Based on the state augmentation approach and the minimum mean square error principle, the optimal filtering problem was investigated in [33] for a class of discrete stochastic systems with finitestep autocorrelated process noises, random one-step sensor delay and missing measurements. In [34], the optimal filter was constructed for stochastic systems subject to the random two-step sensor delays and multiplicative noises. Moreover, an optimal linear filter was constructed in the sense of the minimum mean-square error in [35] for discrete-time systems with measurement-delay and packet dropout. By utilizing the Riccati-like difference equation approach, new recursive state estimation method was developed in [36] for time-varying complex networks subject to missing measurements. However, to the best of authors’ knowledge, these investigations do not provide much attention to the problem of the particle filtering algorithm for discrete nonlinear stochastic systems with random one-step sensor delay and missing measurements.
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Motivated by the above discussions, in this paper, we aim to investigate the particle filtering problem for a class of networked nonlinear systems with random onestep sensor delay and missing measurements. Two typical network-induced phenomena, namely the sensor delay and measurement loss, are taken into account and incorporated into the observation equation. The phenomena of random one-step sensor delay and missing measurements are described by two Bernoulli distributed random variables with known conditional probabilities. Based on the Bayesian filtering framework, the particles and their weights are updated due to taking the random one-step sensor delay and missing measurements into account. Here, by the assumption of first-order Markovian process, we make the first attempt to propose the particle filter for nonlinear systems subject to random one-
In many practical systems, the phenomena of random time-delay and missing measurements are inevitably encountered and the accuracy of the filtering algorithm is degraded [20–25]. Hence, both the time-delay and
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step sensor delay and missing measurements. It is shown that, based on the new particle filtering algorithm and Matlab software, we can recursively compute the state estimate and the error covariance matrices. Finally, we give a numerical example to verify the performance of the proposed filtering algorithm.
Prob {λk = 1} = E {λk } = α,
Prob {λk = 0} = 1 − E {λk } = 1 − α, Prob {ξk = 1} = E {ξk } = β,
Prob {ξk = 0} = 1 − E {ξk } = 1 − β,
Notation. The symbols used in the paper are standard. Rn denotes the n-dimensional Euclidean space. AT represents the transpose of a matrix A. E{x} is the expectation of the random variable x. The identity matrix and the zero matrix are expressed by I and 0 with appropriate dimensions, respectively. Xk = {xj }kj=1 is the set of the state vectors. Zk = {zj }kj=1 stands for the set of the actual output. Yk = {yj }kj=1 is the set of the measured output. N is the number of the particles. δk−j is the Kronecker delta function. If k = j, then δk−j = 1; R otherwise, δk−j = 0. is the integral operation. x ∼ p(x) denotes that the variable x obeys a distribution with probability density function p(x). If a stochastic process {x}∞ k=0 satisfies first-order Markovian property, then p(xk |xk−1 , xk−2 , · · · , x0 ) = p(xk |xk−1 ). If the dimensions of the matrices are not definitely stated, they are considered to be well-matched for algebraic operations.
where α, β ∈ [0, 1] are known scalars and λk , ξk are uncorrelated with other noise signals.
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Remark 1 In the model (3), if λk = 1, yk = zk , it stands for that the sensor receives the data at time instant k successfully; if λk = 0 and ξk = 1, yk = zk−1 , it means that there exists one-step delay; if λk = 0 and ξk = 0, yk = νk , it represents that current measurements are missing completely.
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In order to make the subsequent proof process simple, we make the following three assumptions as in the reference [15].
p (xk |xk−1 , H ) = p (xk |xk−1 ) ,
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Assumption 1 The state xk obeys the following firstorder Markovian process:
Problem Formulation and Preliminaries
xk+1 = fk (xk ) + ωk
Assumption 2 The measured output of the sensor yk and the states xk , xk−1 obey the following relationship p (yk |xk , xk−1 , H ) = p (yk |xk , xk−1 ) . Assumption 3 The previous state Xk−1 and the output of the sensor yk are uncorrelated in terms of proposal density function q(·), i.e., q(Xk−1 |Yk ) = q(Xk−1 |Yk−1 ).
(1) (2)
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zk = hk (xk ) + νk
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Consider the following discrete nonlinear stochastic systems with random sensor delay and missing measurements:
where H is an arbitrary subset of {x0 , Xk−2 , Yk−1 }.
yk = λk zk + (1 − λk )ξk zk−1 + (1 − λk )(1 − ξk )νk (3)
The purpose of this paper is to design a new particle filter in Bayesian filtering framework based on the observation sequence {y1 , y2 , · · · , yk } and above three assumptions.
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where xk ∈ Rn is the state vector, zk ∈ Rm is the actual output, and yk ∈ Rm is the measured output of the sensor. fk (·) : Rn → Rn and hk (·) : Rn → Rm are known nonlinear functions. ωk ∈ Rn and νk ∈ Rm are uncorrelated white noises obeying the arbitrary probability density functions.
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Design of Estimation Algorithm
Due to the phenomena of random one-step sensor delay and missing measurements, the selection of importance weights are different from traditional particle filter. Hence, we simply introduce the importance sampling methods in [37]. Firstly, the following lemma is
The random variables λk and ξk , which describe the phenomena of random one-step sensor delay and missing measurements, satisfy the Bernoulli distribution and
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given in [38] to introduce Baysian theorem and Baysian rule. Lemma 1 Baysian theorem relates joint probability density function to conditional and marginal probability density functions, i.e., two events X and Y have the following relationship:
wki =
p(Xki , Yk ) , q(Xki |Yk )
w ¯ki =
It is easy to see that we can approximate posterior distribution p(Xk |Yk ) from (3) and (6) as follows:
Baysian rule is expressed as: p(Xk |Yk ) ≈ pˆ(Xk |Yk ) =
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1 N
N P
i=1
= wki
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w ¯ki g(Xki ),
+(1 − α)βpνk−1 (yk − hk−1 (xik−1 )) +(1 − α)(1 − β)pνk−1 (yk ),
q(Xk |Yk ) = q(xk |Xk−1 , Yk )q(Xk−1 |Yk )
= q(xk |Xk−1 , Yk )q(Xk−1 |Yk−1 ).
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gˆ(Xk ) ≈
p(yk |xik , xik−1 ) = αpνk (yk − hk (xik ))
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Proof : By applying the assumption 3 and the Bayesian theorem, one has
p(Xk ,Yk ) q(Xk |Yk ) q(Xk |Yk )dXk . p(Xk ,Yk ) q(Xk |Yk ) q(Xk |Yk )dXk
wki g(Xki )
(9)
xik is obtained from the proposal density function q(xk |Xk−1 , Yk ), pνk (·) and pνk−1 (·) stand for the probability density functions of observation noises νk and νk−1 .
(5)
k ,Yk ) Let wk = p(X q(Xk |Yk ) , where wk is the importance weight. Then, by choosing independent and identically N distributed sample Xki i=1 from the proposal density function q(Xk |Yk ), we have:
i=1
p(yk |xik , xik−1 )p(xik |xik−1 ) i wk−1 , i q(xik |Xk−1 , Yk )
where
The following equation can be derived by substituting (5) into (4):
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Theorem 1 Based on the assumptions 1-3, the importance weights wki can be computed recursively as follows: wki =
From the Lemma 1, we can obtain
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gˆ(Xk ) = E [g(Xk )|Yk ] Z = g(Xk )p(Xk |Yk )dXk Z p(Xk |Yk ) = g(Xk ) q(Xk |Yk )dXk . q(Xk |Yk )
g(Xk ) R
w ¯ki δ Xk − Xki .
By applying the importance sampling approach, a recursive computation formula of importance weights wki can be obtained for nonlinear discrete stochastic systems with time-delay and missing measurements.
Based on the Lemma 1 and the proposal density function q(Xk |Yk ), the following optimal estimation of arbitrary function g(Xk ) in the mean square sense can be derived easily:
gˆ(Xk ) =
N X
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p(Y |X)p(X) . p(X|Y ) = p(Y )
Z
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p(X, Y ) = p(X|Y )p(Y ) = p(Y |X)p(X).
p(Xk , Yk ) p(Xk |Yk ) = p(Yk ) p(Xk , Yk ) =R p(Xk , Yk )dXk p(Xk , Yk ) = R p(X ,Y ) . k k q(Xk |Yk ) q(Xk |Yk )dXk
wki . N P wki
(11)
i Hence, we can get xik and Xk−1 from q(xk |Xk−1 , Yk ) and q(Xk−1 |Yk−1 ), respectively. From the Bayesian theorem, the following equation can be obtained:
p(Xk , Yk ) = p(xk , Xk−1 , yk , Yk−1 ) (6)
= p(yk |xk , Xk−1 , Yk−1 )p(xk , Xk−1 , Yk−1 )
i=1
= p(yk |xk , Xk−1 , Yk−1 )
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(12)
Thirdly, also similar to the above two cases, the following equation is computed:
By using the assumptions 1 and 2, the equation (12) can be equivalently rewritten as follows: p(Xk , Yk ) = p(yk |xk , xk−1 )
×p(xk |xk−1 )p(Xk−1 , Yk−1 ).
p(yk |λk = 0, ξk = 0, xk , xk−1 ) = pνk−1 (yk ).
Based on the similar derivation in [15], the likelihood density function p(yk |xk , xk−1 ) can be calculated by using equations (16), (18) and (19) as follows:
(13)
Based on the equations (11) and (13), the importance weights wk can be expressed in the following form:
p(yk |xk , xk−1 ) Z = p(yk , λk |ξk , xk , xk−1 )dλk Z = p(yk |λk , ξk , xk , xk−1 )p(λk |ξk , xk , xk−1 )dλk Z = p(yk |λk , ξk , xk , xk−1 )p(λk )dλk
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p(Xk , Yk ) q(Xk |Yk ) p(yk |xk , xk−1 )p(xk |xk−1 ) p(Xk−1 , Yk−1 ) = q(xk |Xk−1 , Yk ) q(Xk−1 |Yk−1 ) p(yk |xk , xk−1 )p(xk |xk−1 ) = wk−1 . q(xk |Xk−1 , Yk )
(19)
wk =
= αp(yk |λk = 1, xk , xk−1 ) Z +(1 − α) p(yk , ξk |λk = 0, xk , xk−1 )dξk
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= αp(yk |λk = 1, xk , xk−1 ) Z +(1 − α) p(yk |ξk , λk = 0, xk , xk−1 )
Since the output of the sensor yk may be zk−1 or νk if the one-step time-delay or missing measurements happen, the update of importance weights wk in (14) is different from that in the traditional particle filter. Also, the measurement yk depends on the value of the Bernoulli random variables λk and ξk . Since λk and ξk are independent with other signals, we can evaluate the likelihood density as follows:
×p(ξk |λk = 0, xk , xk−1 )dξk
= αp(yk |λk = 1, xk , xk−1 )
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+(1 − α)βp(yk |λk = 0, ξk = 1, xk , xk−1 )
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Firstly, if λk = 1, we can obtain the following equation by (2) and (3):
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(16)
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Remark 2 In the above algorithm, the first attempt is to develop the particle filtering algorithm for a class of networked nonlinear discrete stochastic systems subject to random one-step sensor delay and missing measurements. Due to the phenomena of time delay and missing measurements, the particles and their weights are updated as Theorem 1 to reduce the conservativeness and
(17)
Then, similar to (16), we have p(yk |λk = 0, ξk = 1, xk , xk−1 )
= pνk−1 (yk − hk−1 (xk−1 )).
(20)
According to Theorem 1, a new recursive algorithm can be established to obtain the nonlinear filter for the addressed discrete stochastic nonlinear systems with random one-step sensor delay and missing measurements. We can design the particle filter in Theorem 1 as the following algorithm.
Next, if λk = 0 and ξk = 1, it holds that yk = hk (xk−1 ) + νk−1 .
+(1 − α)βpνk−1 (yk − hk−1 (xk−1 ))
From the equations (14) and (20), we have the recursions of importance weights wki as the equations (9) and (10). The proof of this theorem is complete.
(15)
Hence, it is easy to see that yk − hk (xk ) and νk have the same distribution functions if xk is known, i.e. p(yk |λk = 1, xk , xk−1 ) = pνk (yk − hk (xk )).
= αpνk (yk − hk (xk ))
+(1 − α)(1 − β)pνk−1 (yk ).
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yk = hk (xk ) + νk .
+(1 − α)(1 − β)p(yk |λk = 0, ξk = 0, xk , xk−1 )
(18)
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. i density function, i.e. q(xik |Xk−1 , Yk ) = p(xik |xik−1 ). i N {xk }i=1 were the samples which extracted from the importance distribution density function. 2 Calculate the weights of particles by Theorem 1.
. 1 and the equations (9),(10), the weights can be Due to computed as follows:
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dom one-step sensor delay and missing measurements. Hence, the measurement yk depends on the value of the Bernoulli random variables λk and ξk . Then, the derivation of the likelihood density function p(yk |xk , xk−1 ) is divided into two parts. When the random variable λk = 1, the sensor output yk depends on xk ; otherwise, the output yk depends on ξk , xk and xk−1 . Next, we discuss the random variable ξk taking the value of 0 or 1. If ξk = 1, yk depends on xk−1 ; otherwise, ys depends on νk . Therefore, the key of calculating the importance weights wk is the derivation of the likelihood density function p(yk |xk , xk−1 ) and we made great efforts to address this issue.
wki = αpνk (yk − hk (xik ))
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+(1 − α)βpνk−1 (yk − hk−1 (xik−1 )) i +(1 − α)(1 − β)pνk−1 (yk ) wk−1 .
3 Normalize the weights:
. w ¯ki =
wki . N P i wk
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w ¯ki xik ,
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Step 3. Re-sample: Based on the technology of the sequential importance sampling, the particles are re-sampled. Step 4. Compute the state estimation: x ˆk =
i=1
i=1
w ¯ki (xik − x ˆk )(xik − x ˆ k )T .
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Remark 4 It’s well known that the extended Kalman filter is designed by using the first-order linearization for nonlinear stochastic systems, where the error will lead to the divergence of the proposed filtering algorithm. Hence, the unscented Kalman filtering is proposed based on applying the unscented transformation approach to reduce the error of the the extended Kalman filtering algorithm. However, in many practice systems, the non-Gaussian problems can not be solved by employing the unscented Kalman filtering. Therefore, the particle filtering algorithm, also called the Bayesian estimation algorithm, has been received considerable attention. Simply speaking, the particle filtering algorithm is to approximate the probability density function of the system by employing a set of random samples which spread in the sate space to obtain the minimum variance distribution of the state. Here, the samples or particles, when the number of particles N → ∞, can approximate any form of probability density distribution. What’s more, the particle filtering algorithm isn’t subject to the Gaussian distribution and it can approximate the more wide distribution. Due to the advantage of the particle filtering algorithm for the nonlinear non-Gaussian systems, the particle filtering algorithm proposed in the paper has extensive application than the traditional extended Kalman filter and unscented Kalman filter.
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Step 5. Return to the Step 2. The end.
ensure the accuracy of the state estimation algorithm. Remark 3 In this paper, we consider the phenomena of missing measurements and random one-step sensor delay together when designing the particle filtering algorithm. To be more specific, two Bernoulli distributed random variables with known conditional probabilities (i.e. λk and ξk ) are employed to describe the phenomena of ran-
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An Illustrative Example
We give a simulation example to illustrate the performance of the developed filtering algorithm in this section.
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1.4 e1k in this paper e1k in [15] e1k in [37]
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xk+1 = 3.5xk (1 − xk ) + ωk
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zk = cos(xk π) + x2k + νk
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yk = λk zk + (1 − λk )ξk zk−1 + (1 − λk )(1 − ξk )νk
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x1k x2k x3k
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. The process noises ωk and
0.2
observation noises νk are uncorrelated white noises and satisfy ωk ∼ N (0, Qk ) and νk ∼ N (0, Rk ) with the positive definite covariance matrices Qk = 0.1I3 , Rk = 0.01I3 . Let T x0 = 0.1 0.15 0.2 ; P0|0 = I3 ;
α = 0.8;
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e2k in this paper e2k in [15] e2k in [37]
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and eik denote the estimation error of xik , i.e., eik = xik − x ˆik|k , where i = 1, 2, 3.
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According to Theorem 1, the particle filtering algorithm can be designed based on the Bayesian filtering framework and the assumption of first-order Markovian process. The values of states and their estimates at every time step can be recursively computed by utilizing the given algorithm and Matlab software. The state estimation errors eik (i = 1, 2, 3) of the proposed filtering algorithm are plotted in Figs. 1-3. Figs. 4-6 plot the actual system states xik and their estimates x ˆik|k (i = 1, 2, 3). From the simulations, we can see that the range of error fluctuation in our paper is relatively small compared with the error in the references [15, 37]. It is shown that the proposed filter can estimate the system state well.
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Fig. 2. The error of state x2k
compute the weights of particles according to the equations (9) and (10) in the particle filtering algorithm to reduce the errors in the sampling process.
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In our paper, we analyze and compare the different rates of time-delay and missing measurements in order to investigate the effects from the considered phenomena in depth (i.e., Case I: α = 0.8, β = 0.85, 0.55, 0.25; Case II: α = 0.8, 0.5, 0.2, β = 0.85 ). The corresponding simulation results are given in Figs. 7-12. Based on the simulation results, we can conclude that the different probabilities of sensor delay or missing measurements have a significant influence on the accuracy of the filter. Comparatively speaking, the proposed filter can estimate the system states effectively compared with traditional particle filtering algorithm in [37]. The reason is that we
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Conclusions
The problem of particle filtering problem has been investigated for a class of networked nonlinear stochastic systems subject to random one-step sensor delay and missing measurements. Based on the Bayesian filtering framework and the assumption of first-order Markovian process, the particles and their weights have been updated and a new particle filtering algorithm has been
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1.1 e3k in this paper e3k in [15] e3k in [37]
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Actual state x2k filter in this paper
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Fig. 3. The error of state x3k
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Fig. 5. The trajectories of x2k and x ˆ2k|k
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Fig. 4. The trajectories of x1k and x ˆ1k|k
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Acknowledgments This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants 61174001, 61321062, 11301118 and 11271103.
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proposed. Accordingly, a recursive algorithm has been given to design the nonlinear filter. Moreover, a simulation example has been used to show the feasibility and usefulness of the proposed approach compared with the traditional particle filtering algorithm. Further research topics include the extensions of the obtained results to deal with the particle filtering problems for networked systems with multi-step sensor delay and disturbances as in [39]. The corresponding results will appear in the near future.
References ¨ [1] E. Ozkan, N. Wahlstr¨ om, and S. Godsill, Rao-blackwellised particle filter for star-convex extended target tracking models, 19th International Conference on Information Fusion, (2016) 1193–1199. [2] F. Wang, Y. Zhen, B. Zhong, and R. Ji, Robust infrared target tracking based on particle filter with embedded saliency detection, Information Sciences, 301 (2015) 215–226.
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Actual state x1k β=0.85 β=0.55 β=0.25
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Fig. 7. The trajectories of x1k (α = 0.8)
Fig. 9. The trajectories of x3k (α = 0.8)
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Fig. 8. The trajectories of x2k (α = 0.8)
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Long Xu received the B.Sc. degree in Applied Mathematics from Harbin University, Harbin, China, in 2011, and M.Sc. degree in Applied Mathematics from Harbin University of Science and Technology, Harbin, China, in 2015. He is currently a Ph.D. candidate with Control Science and Engineering, Harbin Institute of Technology, Harbin, China. His current research interests include nonlinear control and filtering, time-varying systems and stochastic systems.
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Kemao Ma received his Bachelor degree in engineering from the Department of Information and Control Engineering, Xi’an Jiaotong University, China, in 1992, and his Ph.D. degree from Harbin Institute of Technology, China, in 1998. Since 2006, he has been a professor at the Control and Simulation Center, School of Astronautics, Harbin Institute of Technology, China. His current research interest includes nonlinear control, non-smooth analysis and control, robust control and their applications to guidance and control of flight vehicles.
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interests include stochastic control,neural networks, complex networks, nonlinear dynamics, time-delay systems, multi-agent systems, and chaotic dynamics.
Fuad E. Alsaadi received the BSc and MSc degrees in electronic and communication from King AbdulAziz University, Jeddah, Saudi Arabia, in 1996 and 2002. He then received the PhD degree in optical wireless communication systems from the University of Leeds, Leeds, UK, in 2011. Between 1996 and 2005, he worked in Jeddah as a communication instructor in the College of Electronics & Communication. He is currently an assistant professor of the Electrical and Computer Engineering Department within the Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He published widely in the top IEEE communications conferences and journals and has received the Carter award, University of Leeds for the best PhD. He has research interests in optical systems and networks, signal processing, synchronization and systems design.
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Wenshuo Lireceived the B.Eng. degree from Shandong University, Jinan, China, in 2012. He is currently working towards the Ph.D. degree in Beihang University, Beijing, China. From Jan. 2016 to Jan. 2017, he was an exchange Ph.D. student in Brunel University London, UK. His research interests include nonlinear filtering, Monte Carlo methods, anti-disturbance control and filtering, and their applications in aerospace and networked systems.
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Yurong Liu was born in China in 1964. He received his B.Sc. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.Sc. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2001. Dr. Liu is currently a professor with the Department of Mathematics at Yangzhou University, China. He also serves as an Associate Editor of Neurocomputing. So far, he has published more than 70 papers in refereed international journals. His current
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Accepted Manuscript
Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements Long Xu, Kemao Ma, Wenshuo Li, Yurong Liu, Fuad E. Alsaadi PII: DOI: Reference:
S0925-2312(17)31710-1 10.1016/j.neucom.2017.10.059 NEUCOM 19039
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
16 August 2017 12 October 2017 30 October 2017
Please cite this article as: Long Xu, Kemao Ma, Wenshuo Li, Yurong Liu, Fuad E. Alsaadi, Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.10.059
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ACCEPTED MANUSCRIPT
Particle Filtering for Networked Nonlinear Systems Subject to Random One-step Sensor Delay and Missing Measurements Long Xu a , Kemao Ma a,∗ , Wenshuo Li b , Yurong Liu c,d , Fuad E. Alsaadi d Department of Control Science and Engineering, Control and Simulation Center, Harbin Institute of Technology, Harbin 150080, China b
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China c
Department of Mathematics, Yangzhou University, Yangzhou 225002, China
Communication Systems and Networks (CSN) Research Group, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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Abstract
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In this paper, the particle filtering problem is investigated for a class of networked nonlinear systems with random one-step sensor delay and missing measurements. The phenomena of missing measurements and random one-step sensor delay are modeled by two random variables, both obeying the Bernoulli distribution. Here, we derive an explicit expression for the likelihood function when the possible occurrence of one-step sensor delay and measurement loss is taken into consideration. Based on this likelihood function, we propose a novel particle filtering algorithm to treat the nonlinear estimation problem in the simultaneous presence of random sensor delay and measurement loss. Finally, a simulation example is given to illustrate the feasibility and advantages of the proposed filtering scheme compared with traditional particle filtering algorithm.
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Introduction
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Key words: Particle filter; networked systems; random one-step sensor delay; missing measurements; bayesian framework.
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In recent years, the nonlinear filtering problems were extensively considered due to wide applications in many fields such as target tracking, location and parameter estimation [1–6]. It is well known that Kalman first proposed the classical Kalman filter in [7]. Up to now, the Kalman filter is still widely used as a standard framework to solve the linear estimation problem. However, in a practical environment, most of the systems have nonlinear characteristics [8, 9], hence the study of nonlinear filtering problems is of important significance. A well-known method, the Monte Carlo method, was proposed in [10] and used to solve the nonlinear filtering problems. In addition, the sequential importance sam-
Preprint submitted to Automatica
pling algorithm was utilized to estimate the nonlinear system states, where the proposed algorithm takes a set of samples extracted from the proposed distribution to approximate the distribution of the target and each sample has a corresponding weight. Due to considering the problem of sample weight degradation in the sequential importance sampling algorithm, the application of sequential importance sampling algorithm in solving nonlinear filtering problem was very limited at that time. In order to avoid this case, the re-sampling technique was introduced into the Monte Carlo importance sampling process and a new filter was proposed in [11], which can effectively solve the problem of the sample weight degradation. By summarizing the sequential importance sam-
3 November 2017
ACCEPTED MANUSCRIPT pling algorithm, the concept of Sequential Monte Carlo method, namely particle filter, was proposed in [12]. The basic idea of this method is to approximate the posterior probability distribution of the system by employing a set of samples (or particles) in order to estimate the state of the nonlinear system. Subsequently, the particle filtering algorithm has been widely applied in many practical problems.
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Due to the wide application of particle filtering approach, the scholars paid close attention to the developments of particle filtering technique in many different research fields, such as modern signal processing, communication, artificial intelligence, autonomous mobile robot and target tracking. Accordingly, a large number of application results were published concerning on particle filtering algorithm. To mention a few, for the situation of randomly missing measurements in engineering practice, a filtering method was proposed in [13] for nonlinear system based on sequential importance sampling and was applied to online state estimation for nonlinear non-Gauss systems. In [14], a new hybrid filtering algorithm was proposed for a class of nonlinear systems with linear state equation and nonlinear measurement equation based on the approaches of Kalman filter and particle filter. Based on Bayesian filtering framework, a new particle filter was designed in [15] for nonlinear systems with random one-step delayed measurements, where the probability of the time-delay is unknown and is estimated by maximum likelihood criterion. When taking the multiple-step time-delay into account, the nonlinear filtering problem was discussed in [16] for nonlinear systems with multiple-step randomly delayed measurements by applying the idea as in [15]. When the data are measured by multiple sensors, the problem of distributed particle filtering was investigated in [17] for nonlinear tracking over wireless sensor networks. Moreover, a distributed particle filter was constructed in [18] for bearings-only tracking of a target moving on the surface of a sphere and a proof of uniform convergence of distributed particle filtering algorithm was given in [19].
missing measurements have attracted the attention of many scholars and a large number of results were reported [26–30]. Recently, in [31], a linear optimal filtering problem was investigated for a class of stochastic timevarying systems subject to state delay and randomly multiple sensor delays based on the innovative analysis approach and recursive projection formula. By using the same method as in [31], the optimal filter was designed in [32] for a networked stochastic state delay systems with missing measurements. Based on the state augmentation approach and the minimum mean square error principle, the optimal filtering problem was investigated in [33] for a class of discrete stochastic systems with finitestep autocorrelated process noises, random one-step sensor delay and missing measurements. In [34], the optimal filter was constructed for stochastic systems subject to the random two-step sensor delays and multiplicative noises. Moreover, an optimal linear filter was constructed in the sense of the minimum mean-square error in [35] for discrete-time systems with measurement-delay and packet dropout. By utilizing the Riccati-like difference equation approach, new recursive state estimation method was developed in [36] for time-varying complex networks subject to missing measurements. However, to the best of authors’ knowledge, these investigations do not provide much attention to the problem of the particle filtering algorithm for discrete nonlinear stochastic systems with random one-step sensor delay and missing measurements.
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Motivated by the above discussions, in this paper, we aim to investigate the particle filtering problem for a class of networked nonlinear systems with random onestep sensor delay and missing measurements. Two typical network-induced phenomena, namely the sensor delay and measurement loss, are taken into account and incorporated into the observation equation. The phenomena of random one-step sensor delay and missing measurements are described by two Bernoulli distributed random variables with known conditional probabilities. Based on the Bayesian filtering framework, the particles and their weights are updated due to taking the random one-step sensor delay and missing measurements into account. Here, by the assumption of first-order Markovian process, we make the first attempt to propose the particle filter for nonlinear systems subject to random one-
In many practical systems, the phenomena of random time-delay and missing measurements are inevitably encountered and the accuracy of the filtering algorithm is degraded [20–25]. Hence, both the time-delay and
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ACCEPTED MANUSCRIPT have the statistical properties as follows:
step sensor delay and missing measurements. It is shown that, based on the new particle filtering algorithm and Matlab software, we can recursively compute the state estimate and the error covariance matrices. Finally, we give a numerical example to verify the performance of the proposed filtering algorithm.
Prob {λk = 1} = E {λk } = α,
Prob {λk = 0} = 1 − E {λk } = 1 − α, Prob {ξk = 1} = E {ξk } = β,
Prob {ξk = 0} = 1 − E {ξk } = 1 − β,
Notation. The symbols used in the paper are standard. Rn denotes the n-dimensional Euclidean space. AT represents the transpose of a matrix A. E{x} is the expectation of the random variable x. The identity matrix and the zero matrix are expressed by I and 0 with appropriate dimensions, respectively. Xk = {xj }kj=1 is the set of the state vectors. Zk = {zj }kj=1 stands for the set of the actual output. Yk = {yj }kj=1 is the set of the measured output. N is the number of the particles. δk−j is the Kronecker delta function. If k = j, then δk−j = 1; R otherwise, δk−j = 0. is the integral operation. x ∼ p(x) denotes that the variable x obeys a distribution with probability density function p(x). If a stochastic process {x}∞ k=0 satisfies first-order Markovian property, then p(xk |xk−1 , xk−2 , · · · , x0 ) = p(xk |xk−1 ). If the dimensions of the matrices are not definitely stated, they are considered to be well-matched for algebraic operations.
where α, β ∈ [0, 1] are known scalars and λk , ξk are uncorrelated with other noise signals.
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Remark 1 In the model (3), if λk = 1, yk = zk , it stands for that the sensor receives the data at time instant k successfully; if λk = 0 and ξk = 1, yk = zk−1 , it means that there exists one-step delay; if λk = 0 and ξk = 0, yk = νk , it represents that current measurements are missing completely.
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In order to make the subsequent proof process simple, we make the following three assumptions as in the reference [15].
p (xk |xk−1 , H ) = p (xk |xk−1 ) ,
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Assumption 1 The state xk obeys the following firstorder Markovian process:
Problem Formulation and Preliminaries
xk+1 = fk (xk ) + ωk
Assumption 2 The measured output of the sensor yk and the states xk , xk−1 obey the following relationship p (yk |xk , xk−1 , H ) = p (yk |xk , xk−1 ) . Assumption 3 The previous state Xk−1 and the output of the sensor yk are uncorrelated in terms of proposal density function q(·), i.e., q(Xk−1 |Yk ) = q(Xk−1 |Yk−1 ).
(1) (2)
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zk = hk (xk ) + νk
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Consider the following discrete nonlinear stochastic systems with random sensor delay and missing measurements:
where H is an arbitrary subset of {x0 , Xk−2 , Yk−1 }.
yk = λk zk + (1 − λk )ξk zk−1 + (1 − λk )(1 − ξk )νk (3)
The purpose of this paper is to design a new particle filter in Bayesian filtering framework based on the observation sequence {y1 , y2 , · · · , yk } and above three assumptions.
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where xk ∈ Rn is the state vector, zk ∈ Rm is the actual output, and yk ∈ Rm is the measured output of the sensor. fk (·) : Rn → Rn and hk (·) : Rn → Rm are known nonlinear functions. ωk ∈ Rn and νk ∈ Rm are uncorrelated white noises obeying the arbitrary probability density functions.
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Design of Estimation Algorithm
Due to the phenomena of random one-step sensor delay and missing measurements, the selection of importance weights are different from traditional particle filter. Hence, we simply introduce the importance sampling methods in [37]. Firstly, the following lemma is
The random variables λk and ξk , which describe the phenomena of random one-step sensor delay and missing measurements, satisfy the Bernoulli distribution and
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given in [38] to introduce Baysian theorem and Baysian rule. Lemma 1 Baysian theorem relates joint probability density function to conditional and marginal probability density functions, i.e., two events X and Y have the following relationship:
wki =
p(Xki , Yk ) , q(Xki |Yk )
w ¯ki =
It is easy to see that we can approximate posterior distribution p(Xk |Yk ) from (3) and (6) as follows:
Baysian rule is expressed as: p(Xk |Yk ) ≈ pˆ(Xk |Yk ) =
(4)
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1 N
N P
i=1
= wki
N X
w ¯ki g(Xki ),
+(1 − α)βpνk−1 (yk − hk−1 (xik−1 )) +(1 − α)(1 − β)pνk−1 (yk ),
q(Xk |Yk ) = q(xk |Xk−1 , Yk )q(Xk−1 |Yk )
= q(xk |Xk−1 , Yk )q(Xk−1 |Yk−1 ).
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gˆ(Xk ) ≈
p(yk |xik , xik−1 ) = αpνk (yk − hk (xik ))
(10)
Proof : By applying the assumption 3 and the Bayesian theorem, one has
p(Xk ,Yk ) q(Xk |Yk ) q(Xk |Yk )dXk . p(Xk ,Yk ) q(Xk |Yk ) q(Xk |Yk )dXk
wki g(Xki )
(9)
xik is obtained from the proposal density function q(xk |Xk−1 , Yk ), pνk (·) and pνk−1 (·) stand for the probability density functions of observation noises νk and νk−1 .
(5)
k ,Yk ) Let wk = p(X q(Xk |Yk ) , where wk is the importance weight. Then, by choosing independent and identically N distributed sample Xki i=1 from the proposal density function q(Xk |Yk ), we have:
i=1
p(yk |xik , xik−1 )p(xik |xik−1 ) i wk−1 , i q(xik |Xk−1 , Yk )
where
The following equation can be derived by substituting (5) into (4):
N P
(8)
Theorem 1 Based on the assumptions 1-3, the importance weights wki can be computed recursively as follows: wki =
From the Lemma 1, we can obtain
1 N
i=1
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gˆ(Xk ) = E [g(Xk )|Yk ] Z = g(Xk )p(Xk |Yk )dXk Z p(Xk |Yk ) = g(Xk ) q(Xk |Yk )dXk . q(Xk |Yk )
g(Xk ) R
w ¯ki δ Xk − Xki .
By applying the importance sampling approach, a recursive computation formula of importance weights wki can be obtained for nonlinear discrete stochastic systems with time-delay and missing measurements.
Based on the Lemma 1 and the proposal density function q(Xk |Yk ), the following optimal estimation of arbitrary function g(Xk ) in the mean square sense can be derived easily:
gˆ(Xk ) =
N X
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p(Y |X)p(X) . p(X|Y ) = p(Y )
Z
(7)
i=1
p(X, Y ) = p(X|Y )p(Y ) = p(Y |X)p(X).
p(Xk , Yk ) p(Xk |Yk ) = p(Yk ) p(Xk , Yk ) =R p(Xk , Yk )dXk p(Xk , Yk ) = R p(X ,Y ) . k k q(Xk |Yk ) q(Xk |Yk )dXk
wki . N P wki
(11)
i Hence, we can get xik and Xk−1 from q(xk |Xk−1 , Yk ) and q(Xk−1 |Yk−1 ), respectively. From the Bayesian theorem, the following equation can be obtained:
p(Xk , Yk ) = p(xk , Xk−1 , yk , Yk−1 ) (6)
= p(yk |xk , Xk−1 , Yk−1 )p(xk , Xk−1 , Yk−1 )
i=1
= p(yk |xk , Xk−1 , Yk−1 )
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(12)
Thirdly, also similar to the above two cases, the following equation is computed:
By using the assumptions 1 and 2, the equation (12) can be equivalently rewritten as follows: p(Xk , Yk ) = p(yk |xk , xk−1 )
×p(xk |xk−1 )p(Xk−1 , Yk−1 ).
p(yk |λk = 0, ξk = 0, xk , xk−1 ) = pνk−1 (yk ).
Based on the similar derivation in [15], the likelihood density function p(yk |xk , xk−1 ) can be calculated by using equations (16), (18) and (19) as follows:
(13)
Based on the equations (11) and (13), the importance weights wk can be expressed in the following form:
p(yk |xk , xk−1 ) Z = p(yk , λk |ξk , xk , xk−1 )dλk Z = p(yk |λk , ξk , xk , xk−1 )p(λk |ξk , xk , xk−1 )dλk Z = p(yk |λk , ξk , xk , xk−1 )p(λk )dλk
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p(Xk , Yk ) q(Xk |Yk ) p(yk |xk , xk−1 )p(xk |xk−1 ) p(Xk−1 , Yk−1 ) = q(xk |Xk−1 , Yk ) q(Xk−1 |Yk−1 ) p(yk |xk , xk−1 )p(xk |xk−1 ) = wk−1 . q(xk |Xk−1 , Yk )
(19)
wk =
= αp(yk |λk = 1, xk , xk−1 ) Z +(1 − α) p(yk , ξk |λk = 0, xk , xk−1 )dξk
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= αp(yk |λk = 1, xk , xk−1 ) Z +(1 − α) p(yk |ξk , λk = 0, xk , xk−1 )
Since the output of the sensor yk may be zk−1 or νk if the one-step time-delay or missing measurements happen, the update of importance weights wk in (14) is different from that in the traditional particle filter. Also, the measurement yk depends on the value of the Bernoulli random variables λk and ξk . Since λk and ξk are independent with other signals, we can evaluate the likelihood density as follows:
×p(ξk |λk = 0, xk , xk−1 )dξk
= αp(yk |λk = 1, xk , xk−1 )
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+(1 − α)βp(yk |λk = 0, ξk = 1, xk , xk−1 )
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Firstly, if λk = 1, we can obtain the following equation by (2) and (3):
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(16)
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Remark 2 In the above algorithm, the first attempt is to develop the particle filtering algorithm for a class of networked nonlinear discrete stochastic systems subject to random one-step sensor delay and missing measurements. Due to the phenomena of time delay and missing measurements, the particles and their weights are updated as Theorem 1 to reduce the conservativeness and
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Then, similar to (16), we have p(yk |λk = 0, ξk = 1, xk , xk−1 )
= pνk−1 (yk − hk−1 (xk−1 )).
(20)
According to Theorem 1, a new recursive algorithm can be established to obtain the nonlinear filter for the addressed discrete stochastic nonlinear systems with random one-step sensor delay and missing measurements. We can design the particle filter in Theorem 1 as the following algorithm.
Next, if λk = 0 and ξk = 1, it holds that yk = hk (xk−1 ) + νk−1 .
+(1 − α)βpνk−1 (yk − hk−1 (xk−1 ))
From the equations (14) and (20), we have the recursions of importance weights wki as the equations (9) and (10). The proof of this theorem is complete.
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Hence, it is easy to see that yk − hk (xk ) and νk have the same distribution functions if xk is known, i.e. p(yk |λk = 1, xk , xk−1 ) = pνk (yk − hk (xk )).
= αpνk (yk − hk (xk ))
+(1 − α)(1 − β)pνk−1 (yk ).
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yk = hk (xk ) + νk .
+(1 − α)(1 − β)p(yk |λk = 0, ξk = 0, xk , xk−1 )
(18)
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ACCEPTED MANUSCRIPT Algorithm 1 The steps of the particle filtering algorithm are shown as follows:: Step 1. Initialisation: The particles are extracted from the prior density function p(x0 ) to obtain a new particle set {xi0 }N i=1 . The weight of all particles is N1 . For k = 1, 2, · · · Step 2. Sequential importance re-sampling: 1 Choose the prior density function as the proposal
. i density function, i.e. q(xik |Xk−1 , Yk ) = p(xik |xik−1 ). i N {xk }i=1 were the samples which extracted from the importance distribution density function. 2 Calculate the weights of particles by Theorem 1.
. 1 and the equations (9),(10), the weights can be Due to computed as follows:
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dom one-step sensor delay and missing measurements. Hence, the measurement yk depends on the value of the Bernoulli random variables λk and ξk . Then, the derivation of the likelihood density function p(yk |xk , xk−1 ) is divided into two parts. When the random variable λk = 1, the sensor output yk depends on xk ; otherwise, the output yk depends on ξk , xk and xk−1 . Next, we discuss the random variable ξk taking the value of 0 or 1. If ξk = 1, yk depends on xk−1 ; otherwise, ys depends on νk . Therefore, the key of calculating the importance weights wk is the derivation of the likelihood density function p(yk |xk , xk−1 ) and we made great efforts to address this issue.
wki = αpνk (yk − hk (xik ))
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+(1 − α)βpνk−1 (yk − hk−1 (xik−1 )) i +(1 − α)(1 − β)pνk−1 (yk ) wk−1 .
3 Normalize the weights:
. w ¯ki =
wki . N P i wk
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i=1
w ¯ki xik ,
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N X
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Step 3. Re-sample: Based on the technology of the sequential importance sampling, the particles are re-sampled. Step 4. Compute the state estimation: x ˆk =
i=1
i=1
w ¯ki (xik − x ˆk )(xik − x ˆ k )T .
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Pk =
N X
Remark 4 It’s well known that the extended Kalman filter is designed by using the first-order linearization for nonlinear stochastic systems, where the error will lead to the divergence of the proposed filtering algorithm. Hence, the unscented Kalman filtering is proposed based on applying the unscented transformation approach to reduce the error of the the extended Kalman filtering algorithm. However, in many practice systems, the non-Gaussian problems can not be solved by employing the unscented Kalman filtering. Therefore, the particle filtering algorithm, also called the Bayesian estimation algorithm, has been received considerable attention. Simply speaking, the particle filtering algorithm is to approximate the probability density function of the system by employing a set of random samples which spread in the sate space to obtain the minimum variance distribution of the state. Here, the samples or particles, when the number of particles N → ∞, can approximate any form of probability density distribution. What’s more, the particle filtering algorithm isn’t subject to the Gaussian distribution and it can approximate the more wide distribution. Due to the advantage of the particle filtering algorithm for the nonlinear non-Gaussian systems, the particle filtering algorithm proposed in the paper has extensive application than the traditional extended Kalman filter and unscented Kalman filter.
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Step 5. Return to the Step 2. The end.
ensure the accuracy of the state estimation algorithm. Remark 3 In this paper, we consider the phenomena of missing measurements and random one-step sensor delay together when designing the particle filtering algorithm. To be more specific, two Bernoulli distributed random variables with known conditional probabilities (i.e. λk and ξk ) are employed to describe the phenomena of ran-
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An Illustrative Example
We give a simulation example to illustrate the performance of the developed filtering algorithm in this section.
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ACCEPTED MANUSCRIPT Consider the following nonlinear discrete stochastic systems:
1.4 e1k in this paper e1k in [15] e1k in [37]
1.2
xk+1 = 3.5xk (1 − xk ) + ωk
1
zk = cos(xk π) + x2k + νk
0.8
yk = λk zk + (1 − λk )ξk zk−1 + (1 − λk )(1 − ξk )νk
0.6
x1k x2k x3k
T
0.4
. The process noises ωk and
0.2
observation noises νk are uncorrelated white noises and satisfy ωk ∼ N (0, Qk ) and νk ∼ N (0, Rk ) with the positive definite covariance matrices Qk = 0.1I3 , Rk = 0.01I3 . Let T x0 = 0.1 0.15 0.2 ; P0|0 = I3 ;
α = 0.8;
0 −0.2 −0.4
0
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where xk =
10
20
30
40
50
k/time step
x ˆ0|0 = β = 0.85
−0.9 −0.85 −0.8
T
Fig. 1. The error of state x1k
;
e2k in this paper e2k in [15] e2k in [37]
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1.4
and eik denote the estimation error of xik , i.e., eik = xik − x ˆik|k , where i = 1, 2, 3.
1.2 1
According to Theorem 1, the particle filtering algorithm can be designed based on the Bayesian filtering framework and the assumption of first-order Markovian process. The values of states and their estimates at every time step can be recursively computed by utilizing the given algorithm and Matlab software. The state estimation errors eik (i = 1, 2, 3) of the proposed filtering algorithm are plotted in Figs. 1-3. Figs. 4-6 plot the actual system states xik and their estimates x ˆik|k (i = 1, 2, 3). From the simulations, we can see that the range of error fluctuation in our paper is relatively small compared with the error in the references [15, 37]. It is shown that the proposed filter can estimate the system state well.
0.8 0.6 0.4
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0
10
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50
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Fig. 2. The error of state x2k
compute the weights of particles according to the equations (9) and (10) in the particle filtering algorithm to reduce the errors in the sampling process.
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In our paper, we analyze and compare the different rates of time-delay and missing measurements in order to investigate the effects from the considered phenomena in depth (i.e., Case I: α = 0.8, β = 0.85, 0.55, 0.25; Case II: α = 0.8, 0.5, 0.2, β = 0.85 ). The corresponding simulation results are given in Figs. 7-12. Based on the simulation results, we can conclude that the different probabilities of sensor delay or missing measurements have a significant influence on the accuracy of the filter. Comparatively speaking, the proposed filter can estimate the system states effectively compared with traditional particle filtering algorithm in [37]. The reason is that we
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5
Conclusions
The problem of particle filtering problem has been investigated for a class of networked nonlinear stochastic systems subject to random one-step sensor delay and missing measurements. Based on the Bayesian filtering framework and the assumption of first-order Markovian process, the particles and their weights have been updated and a new particle filtering algorithm has been
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1.1 e3k in this paper e3k in [15] e3k in [37]
1.2
Actual state x2k filter in this paper
1
1
0.9
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10
Fig. 3. The error of state x3k
30
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Fig. 5. The trajectories of x2k and x ˆ2k|k
Actual state x3k filter in this paper
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1.1 Actual state x1k filter in this paper
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k/time step
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0
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k/time step 50
ˆ3k|k Fig. 6. The trajectories of x3k and x
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0.5
Fig. 4. The trajectories of x1k and x ˆ1k|k
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Acknowledgments This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grants 61174001, 61321062, 11301118 and 11271103.
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proposed. Accordingly, a recursive algorithm has been given to design the nonlinear filter. Moreover, a simulation example has been used to show the feasibility and usefulness of the proposed approach compared with the traditional particle filtering algorithm. Further research topics include the extensions of the obtained results to deal with the particle filtering problems for networked systems with multi-step sensor delay and disturbances as in [39]. The corresponding results will appear in the near future.
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Kemao Ma received his Bachelor degree in engineering from the Department of Information and Control Engineering, Xi’an Jiaotong University, China, in 1992, and his Ph.D. degree from Harbin Institute of Technology, China, in 1998. Since 2006, he has been a professor at the Control and Simulation Center, School of Astronautics, Harbin Institute of Technology, China. His current research interest includes nonlinear control, non-smooth analysis and control, robust control and their applications to guidance and control of flight vehicles.
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interests include stochastic control,neural networks, complex networks, nonlinear dynamics, time-delay systems, multi-agent systems, and chaotic dynamics.
Fuad E. Alsaadi received the BSc and MSc degrees in electronic and communication from King AbdulAziz University, Jeddah, Saudi Arabia, in 1996 and 2002. He then received the PhD degree in optical wireless communication systems from the University of Leeds, Leeds, UK, in 2011. Between 1996 and 2005, he worked in Jeddah as a communication instructor in the College of Electronics & Communication. He is currently an assistant professor of the Electrical and Computer Engineering Department within the Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He published widely in the top IEEE communications conferences and journals and has received the Carter award, University of Leeds for the best PhD. He has research interests in optical systems and networks, signal processing, synchronization and systems design.
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Wenshuo Lireceived the B.Eng. degree from Shandong University, Jinan, China, in 2012. He is currently working towards the Ph.D. degree in Beihang University, Beijing, China. From Jan. 2016 to Jan. 2017, he was an exchange Ph.D. student in Brunel University London, UK. His research interests include nonlinear filtering, Monte Carlo methods, anti-disturbance control and filtering, and their applications in aerospace and networked systems.
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Yurong Liu was born in China in 1964. He received his B.Sc. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.Sc. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2001. Dr. Liu is currently a professor with the Department of Mathematics at Yangzhou University, China. He also serves as an Associate Editor of Neurocomputing. So far, he has published more than 70 papers in refereed international journals. His current
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