Stochastic stability of modified extended Kalman filter over fading channels with transmission failure and signal fluctuation

Signal Processing 138 (2017) 220–232

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Stochastic stability of modified extended Kalman filter over fading channels with transmission failure and signal fluctuation Xiangdong Liu a,b, Luyu Li a,b, Zhen Li a,b,∗, Herbert H.C. Iu c, Tyrone Fernando c a

School of Automation, Beijing Institute of Technology, Beijing 100081, China Key Laboratory for Intelligent Control & Decision on Complex Systems, Beijing Institute of Technology, Beijing 100081, China c School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia b

a r t i c l e

i n f o

Article history: Received 12 August 2016 Revised 21 January 2017 Accepted 27 March 2017 Available online 29 March 2017 Keywords: Extended Kalman filter Stochastic stability Fading channel Transmission failure Signal fluctuation

a b s t r a c t The observations of nonlinear systems, exposed to a fading channel, greatly suffer from both transmission failure and signal fluctuation. This paper focuses on the design-oriented analysis of nonlinear estimator based on a modified extended Kalman filter (MEKF) over fading wireless networks. Bernoulli process and Rayleigh fading are taken into consideration to model transmission failure and signal fluctuation, respectively. The offline sufficient conditions are established for the boundedness of the expectations of the prediction error covariance matrices sequence (PECMS) of the MEKF, which shows the existence of a crucial arrival rate. Furthermore, based on the derived upper bound of PECMS, further sufficient conditions are provided for mean-square bounded estimate error of the MEKF using the fixed-point theorem. Numerical examples are also given to verify the analytical results and demonstrate the feasibility of the proposed methods. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Over the past decade, wireless sensors and actuators have received a lot of attention due to their advantages of low-costs and easy of expansion [1,2]. Modern industrial systems are widely equipped with wireless sensors, bringing high requirements for the monitoring systems [3–6]. The most effective way to realize the monitoring is by means of state estimation using linear Kalman filter (KF) and nonlinear filter, i.e., Kalman variants, which catalyzes the development of nonlinear filter because the nonlinear system is the overwhelming majority in practice [7–12]. The stochastic stability of the filter is a necessary condition to guarantee the effectiveness of monitoring systems. However, the communication channel between wireless sets are susceptible to environmental influence. Thus, the drawback due to the wireless communication channels must be taken into consideration when conducting the analysis of the stochastic stability and performance of the filter. The major constraints of wireless channel, reducing the estimation performance, are the transmission bandwidth and fading channel [13]. On one hand, the filter under the band-limited channel is confronted with the challenges of

∗

Corresponding author. [email protected] E-mail addresses: [email protected] (X. Liu), [email protected] (L. Li), [email protected] (Z. Li), [email protected] (H.H.C. Iu), tyrone.fernando@ uwa.edu.au (T. Fernando). http://dx.doi.org/10.1016/j.sigpro.2017.03.027 0165-1684/© 2017 Elsevier B.V. All rights reserved.

the time-delay and quantization effect, which has been profoundly studied by Shi et al. [14], Wu and Wang [15], Su et al. [16,17] and Caballero-Águila et al. [18]. On the other hand, the fading channel also causes unstable issues to the filter so that the stochastic stability and performance analysis of filter under fading channel becomes indispensable for the estimation systems design [19,20]. The crucial factor for an effective analysis of practical estimator is the modeling precision of the fading channel. Some research utilizes a binary treatment for the receiving information through the fading channel, which either trusts the information and utilizes it as an observation or drops it as a transmission failure. Such filtering under that channel structure was named as the filter with intermittent observations, and the stochastic stability of KF with intermittent observations for linear time invariant (LTI) was firstly studied by Sinopoli et al. [21]. That work pointed out that the prediction error covariance matrices sequence (PECMS) of the filter was random rather than deterministic, and the expectation of PECMS was exponentially bounded if the arrival rate exceeded a critical probability when the arrival of the observations conformed to a Bernoulli process. By utilizing more complicated channel model to describe the transmission failure, i.e., the Gilbert-Elliott channel model and finite state Markov process, a variety of research extended the analysis of filter with intermittent observations to more general application scenario [22–25]. Moreover, some research extended the work from LTI system to nonlinear system [26–29].

X. Liu et al. / Signal Processing 138 (2017) 220–232

The other modeling method of fading channel is to describe the effectiveness of the observation information by the signal to noise ratio (SNR), which is also named as signal fluctuation [19,30,31]. It was pointed out that PECMS of the filter was a random variable because the signal fluctuation introduced randomness into channels, and upper bounds of means of PECMS was deduced if the channel’s SNR followed the specified distribution [30]. Besides, Quevedo further proposed that SNR was highly related to the channel gain, which was determined by the engineering parameters, i.e., the bit-rate and power level [19,31]. Furthermore, the performance analysis was extended from the filter level to the whole wireless estimation system level in [19,31,32]. Among all these works, the filter performance was analyzed in [33] and [34] under Rayleigh fading channel by verifying the upper error outage probability, which was of practical importance. Because both the modeling methods for fading channel are reasonable, an unified consideration about both the transmission failure and signal fluctuation is able to handle more comprehensive problems introduced by the practical problem from channels [20,35]. KF for LTI system with both transmission failure and signal fluctuation was taken into consideration in [35], where the sufficient and necessary conditions for the stochastic boundedness of PECMS were put forward by the modified Lyapunov and Riccati iteration methods, respectively. The work was further extended to the time-varying KF with more generated fading model, where the transmission failure of channel was described as a Markov chain [20]. In the case of the nonlinear system, an UKF based filter with both disturbances was studied in [36]. Similar to [35], the mean convergence of the PECMS was studied and an upper bound sequence for the PECMS of UKF was given. However, PECMS is a significant criterion of KF for linear systems because the estimation error is a zero mean Gaussian vector with the covariance matrix equal to the PECMS. On the contrary, it becomes unsuitable for nonlinear system only in terms of the stability and performance of PECMS so that the mean-square estimation error is the proper indicator for nonlinear system. Moreover, the unknown diagonal matrix similar to [27], which was a part of the parameters to calculate the upper bound, made the theorem in [36] difficult in application as an off-line analysis method for general nonlinear system. Motivated by these concerns for nonlinear systems, it is of significant necessity to study off-line sufficient conditions for the stochastic boundedness of PECMS and estimate error with both transmission failure and signal fluctuation. This effort can be utilized to design and analyze the fusion estimator over fading wireless networks. This paper focuses on the MEKF over fading channel with both disturbances and off-line sufficient conditions are established for the boundedness of both the mean of PECMS and the mean-square of the estimate error. Because the sufficient conditions for the boundedness of estimate error contain the relationship between the upper mean bound of PECMS and the system Jacobi matrix, an upper bound sequence for the mean of PECMS is also proposed in this paper. The rest of this paper is organized as follows. Section 2 introduces the nonlinear system and the fading channel which the observations are transmitted through. Moreover, the MEKF is established based on EKF and a proposed drop strategy. In Section 3, it is proved that there exists a critical value λc . If the arrival rate λ˜ > λc is guaranteed, the mean of the PECMS (i.e., E[Pˆt+1|t ]) will be bounded for all initial conditions. In Section 4, an explicit expression sequence is proposed as the upper bound of the PECMS. Section 5 further derives the sufficient off-line conditions for the boundedness of et |t −1  based on the upper bound of the PECMS. Section 6 conducted various numerical simulations to verify the theorems in previous sections.

221

The following standard notations are adopted throughout this paper. The norm of vector x stands for the Euclidian norm, and the norm of matrix A stands for the spectral norm. E(x ) denotes the expectation value of x, and the E(x|y ) denotes the expectation value of x conditional on y. In stands for the identity matrix with dimension n, and the I stands for the identity matrix with the suitable dimension. The matrix [A01

0 ] A2

is shortened as A1 A2 .

Finally, x ∼ N (x¯, P ) express that x follow the Gaussian distribution with x¯ mean and P covariance. 2. Problem statement Consider the discrete time nonlinear dynamical system:

xt+1 = f (xt ) + ωt , zt = h(xt ) + νt ,

(1)

where xt ∈ Rn is the state and zt ∈ R p is the measured output. The system function f(x) and estimate function h(x) are continuously differentiable at every x. The process noise ωt ∈ Rn and measurement noise νt ∈ R p are both white Gaussian noise with the covariance matrices Q > 0 and R > 0, respectively. It is assumed that the initial state x0 is also Gaussian random vector with the covariance matrix R0 . Moreover, ωt ,ν t and x0 are independent with each other. The measurement zt is transmitted over a wireless fading channel with both fluctuant and transmission failure. 2.1. Effects of channel fading with transmission failure and signal fluctuation In this part, the impact of a time-varying fading communication channel will be modeled on the observation. Let zt and zt represent the measurement in system (1) and the received observation of filter, respectively. The model of fading channel with both fluctuation and transmission failure is thus given by Xiao et al. [35]:

zt = ξt zt + ηt ,

(2)

where ηt ∈ R p is the channel additive noise, which is white Gaussian noise with covariance matrices  > 0. ξt ∈ R represents the fading channel, which consists of transmission failure and gain fluctuation, i.e.,

ξt = γt ϑt .

(3)

The change gain ϑt is caused by the fluctuation, whose most common statistical model is Rayleigh fading. If ιt = ϑt2 , by the property of Rayleigh fading with the parameter  , ιt is white and its distribution is that, ιt ∼  exp(−ιt ). The arrival of the observation at time t is defined as a binary random γ t :



γt =

1 0

the filter successfully get the observation the observation suffers from the transmission failure. (4)

γ t is a Bernoulli process with the parameter λ, which means that γ t is a sequence of independent identically distributed with the arrival rate P{γt = 1} = λ [21]. The observation function of discrete-time nonlinear dynamical system together with the time-varying fading communication channel can be written as:

zt =

γt ϑt h(xt ) + γt ϑt νt + ηt ,

(5)

where x0 , ωt , ν t , ηt , γ t and ϑt are uncorrelated with each other. Remark 1. By the help of time-stamped technology, the information of γ t together with the observation zt are available for the filter at time t. It is assumed that the channel gain ϑt is valid for the filter at time t by the wireless communication technology in [37]. Also, it could be supposed that the channel gain remains constant during the transfer of the tth data, which is suitable when

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X. Liu et al. / Signal Processing 138 (2017) 220–232

the data is transferred through a low bandwidth communication channel [30]. 2.2. Drop strategy for observation over fading channel

xˆt |t = xˆt |t −1 + γ˜t Kt (yt − h(xˆt |t −1 )),

As already discussed in [38], the estimation over fading channel using a suboptimal estimator needs a drop strategy to optimize the performance of the filtering. Because the EKF is a suboptimal estimator for the system mentioned in Section 2.1, a drop strategy is required to balance the information loss and communication noise. A gate β G is chosen in order to drop the measurements with small gain fluctuation, i.e., |ϑt | ≤ β G . The corresponding communication channel under the receiver design strategy thus can be modeled by:

zt = ξ˜t h(xt ) + ξ˜t νt + ηt ,

(6)

where ξ˜t = γ˜t ϑ˜t and γ˜t is



γ˜t =

1 0

γt = 1 and |ϑt | > βG otherwise

.

(7)

Obviously, γ˜t is still a Bernoulli process with

λ˜  P{γ˜t = 1} = λ × P{|ϑt | > βG } = λexp(−βG2 ),

˜ιt ∼  · exp(−˜ιt ) · exp(β ) ,

(9)

for ˜ι > βG2 . Obviously, γ˜t and ϑ˜t are still independent with each other. For convenience, yt = zt /ϑ˜t is defined. Then, the discrete-time nonlinear dynamical system under this drop strategy can be written as:

xt+1 = f (xt ) + ωt , yt = γ˜t h(xt ) + γ˜t νt + ηt /ϑ˜t .

(10)

For convenience, ν˜t = νt + η˜t is further defined. It is easy got ϑt

that ν˜ is also a white Gaussian process with the time-varying covariance matrix R˜t = R +  . ˜2 ϑt

2.3. Modified extended Kalman filter In this section, the MEKF for the nonlinear system together with fading channel as (10) will be constructed. The MEKF is based on the two-step extended Kalman Filter and drop strategy in Section 2.2. For convenience, denote the one step state prediction as xˆt |t −1 , which is the estimate of xt with the knowledge of {y1 , . . . yt−1 ; ϑ1 , . . . , ϑt−1 ; γ1 , . . . γt−1 }. Similarly, define the measurement as xˆt |t , which is the estimate of xt with the knowledge of {y1 , . . . yt ; ϑ1 , . . . , ϑt ; γ1 , . . . γt }. At the same time, denote the error covariance matrix for the prediction and the measurement is defined by Pˆt |t −1 and Pˆt |t . Because the function f(x) and h(x) are continuously differentiable at every x, it can be presented by the Taylor expansion as:

f (xt ) = f (xˆt |t ) + At et |t + φ (xt , xˆt |t ),

h(xt ) = h(xˆt |t −1 ) + Ct et |t −1 + χ (xt , xˆt |t −1 ),

(11)

where the matrix At and Ct are the Jacobi matrices of nonlinear function f and h at xˆt |t and xˆt |t −1 , respectively, i.e.,

  ∂f ∂h  At = , Ct = .  ∂ x x=xˆt |t ∂ x x=xˆt |t −1

Pˆt |t = Pˆt |t −1 − γ˜t Kt Ct Pˆt |t −1 ,

(12)

φ (xt , xˆt |t ) and χ (xt , xˆt |t −1 ) are the second order residuals for the Taylor polynomial of f(x) and h(x), respectively. The MEKF based on the two-step extended Kalman Filter and drop strategy has two steps in one iteration. One of the step is

(13)

where

Kt = Pˆt |t −1CtT (Ct Pˆt |t −1CtT + R˜t )−1 .

(14)

The other step is to calculate the one step prediction:

xˆt+1|t = f (xˆt |t ), Pˆt+1|t = At Pˆt |t AtT + Q.

(15)

Denote the estimation error as et |t −1 = xt − xˆt |t −1 et |t = xt − xˆt |t , respectively. It can be deduced that,

and

et+1|t = f (xt ) + ωt − f (xˆt |t ) = At et |t + ωt + φ (xt , xˆt |t ),

(16)

et |t = xt − xˆt |t −1 − γ˜t Kt (yt − h(xˆt |t −1 ))

= (I − γ˜t Kt Ct )et |t −1 − γ˜t Kt [ν˜t + χ (xt , xˆt |t −1 )].

(8)

˜ is denoted as the filtering arrival rate. Furthermore, where the λ γ˜t = 0 indicates the occurrence of transmission failure. Then, ϑ˜t makes sense only in the case that γ˜t = 1, i.e., |ϑ˜t | > βG . Denote ˜ιt = ϑ˜t2 . In the case that γ˜t = 1, 2 G

the measurement update by using yt , γ˜t and ϑ˜t , which can be easily deduced by zt , γ t and ϑt . The step of measurement update is that,

(17)

By combining Eqs. (16) and (17), the recursion formula of et+1|t can be got such that,

et+1|t = (At − γ˜t At Kt Ct )et |t −1 + mt + st ,

(18)

where

φ (xt , xˆt |t −1 ) − γ˜t At Kt χ (xt , xˆt |t −1 ), st = ωt − γ˜t At Kt ν˜t .

mt =

(19)

3. Boundedness of the error covariance matrices Because it is hard to directly analyze the PECMS of the MEKF, an equivalent method is proposed. Note a system satisfies that

xt+1 = At xt + Q 1/2 ω ˜ t, 1/2 ˜ yt = Ct xt + R ν˜t , t

(20)

where At , Ct , Q, R˜t are defined in Section 2.3 and ω ˜ t and ν˜t are unit white Gaussian noise, and independent with each other. Then, if the one step predict of the Kalman filter for system (20) is denoted as xˆt |t −1 , the covariance of xˆt |t −1 (i.e., Eω˜ i ,ν˜i [(xt − xˆt |t −1 )T (xt − xˆt |t −1 )]) is equivalent to the PECMS of the MEKF by the Eqs. (13) and (15) in Section 2.3. The state transition matrix of the system (20) from the step  l−1 t to t + l is defined as t+l,t = t+ Ai . For the further analysis i=t of the Kalman filter of system (20), the following concepts are provided regarding the observability and detectability. Definition 1. Firstly, similarly like [39], the observability Gramian of the pair (At , Ct ) in the system (20) is defined as

Ot+l,t =

t+l 

Ti,t CiT Ci i,t ,

(21)

i=t

Secondly, the pair (At , Ct ) in the system (20) is said to be uniformly observable if there exists a positive integer l > 0, and the constant real numbers obs, obs > 0 such that for all t ≥ 0, there exists

0 < obsI ≤ Ot+l,t ≤ obsI.

(22)

Finally, the pair (At , Ct ) is said to be uniformly detectable if there exist l > 0 and the constants 0 ≤ decα < 1 and decβ > 0 such that whenever

t+l,t x ≥ decαx,

(23)

there holds

xT Ot+l,t x ≥ decβ xT x.

(24)

X. Liu et al. / Signal Processing 138 (2017) 220–232

It is obvious that the uniform observable implies the uniform detectable. Assumption 1. Pˆ1|0 > 0 and a, c, q, r, q, r > 0, such that:

a ≤ At  ≤ a,

there

exist

the

real

By using (31), it can be obtained that, T ¯ T ˜ ¯ ¯T T E(x¯¯t+ l x¯t+l |x¯t ) = x¯t Tt [ (1 − λ )In1  In2 ]Tt t+l,t t+l,t

numbers

˜ )In  In ]Tt x¯¯t + Ft , × TtT [(1 − λ 1 2

2

  t+l−1

˜2 Ft = Eω˜ i ,ν˜i λ

rI ≤ R ≤ rI,

 ≤ ηI.

 t+l−1

˜ > λc . obs, obs, such that lim E[Pˆt+1|t ] < +∞ for the arrival rate λ t→+∞

Proof. At first, it is jointly considered that the PECMS of MEKF is equal to the covariance of Kalman filter estimation xˆt |t −1 and the fact that the Kalman filter is the optimal filter for time-varying linear systems. Thus, by defining a suboptimal filter for the system (20) as xˇt |t −1 , it is clear that,

E(Pˆt |t −1 ) = E[(xˆt |t −1 − xt )T (xˆt |t −1 − xt )] (26)

To propose a reasonable suboptimal filter, an orthogonal matrix Tt is constructed to transform symmetric observability Gramian Ot+l,t into a diagonal matrix in the decreasing order such that O˜ t+l,t = Tt Ot+l,t TtT . The matrix satisfies that, 1 ˜2 O˜ t+l,t = O˜ t+ l,t  Ot+l,t ,

(27)

2 and O˜ t+ are the diagonal matrices of dimensions l,t 2 1 ˜ obsI ≤ Ot+l,t < decβ I, decβ I ≤ O˜ t+ ≤ obsI. l,t

n1

and n2 , and Then, the suboptimum filter xˇt +l |t +l−1 can be designed in the following way,

xˇt +l |t +l−1 = 

(

)

1 −1 O˜ t+ l,t

 0]Tt

t+ l−1 

+ [0  I2 ]xˇt |t −1 ],

Due to the boundedness of At , Ct , R˜t and Q, Ft therefore has an upper bound F, which is a positive constant, i.e.,

(28)

At the same time, for the first part of (32), a new transition ˜ )In  In ]. Then, the ˆ t+l,t = t+l,t T T [(1 − λ matrix is denoted as  t 1 2 corresponding observability Gramian is

˜ )In  In ]Tt Ot+l,t T T [(1 − λ ˜ )In  In ] Oˆ t+l,t = [(1 − λ t 1 2 1 2 ˜ )In  In ]O˜ t+l,t [(1 − λ ˜ )In  In ]. = [ (1 − λ 1 2 1 2 xT Oˆ t+l,t x < decβ xT x.

Following the converse-negative proposition of the definition of the uniform detectability, it can be deduced that

ˆ t+l,t x < decαx, where 0 ≤ decα < 1. 

yi , if γ˜i = 1 . Ci i,t xˇt |t −1 , if γ˜i = 0

(29)

˜ )In  In ]Tt T x¯¯tT TtT [(1 − λ 1 2 t+l,t ˜ )In  In ]Tt x¯¯t ×t+l,t TtT [(1 − λ 1 2 2

≤ decα x¯¯tT x¯¯t .

2

t→+∞

(30)

+ t+l,t TtT [0  In2 ]Tt (xˇt |t −1 − xt ) − Ft

(

)

 0]Tt

i=t



T T i,t Ci Ei .

(41)

As decα < 1, it is easy to deduce that the first term of (41) tends to 0, and the second term of equation is a bounded constant when t → +∞. Combining the Eq. (26), it can be got that

lim E[Pˆt |t ] < lim E[x¯¯t x¯¯tT ] ≤ lim E[x¯¯tT x¯¯t ]I < +∞.

t→+∞

t→+∞

t→+∞

(42)

By Pˆt+1|t = At Pˆt |t AtT + Q and the boundedness of At and Q, limt→+∞ E[Pˆt+1|t ] < +∞ can be finally concluded.  4. Upper bounds on error covariance matrices for scalar measurement and Rayleigh fading

˜ )t+l,t T T [In  0]Tt (xˇt |t −1 − xt ) x¯¯t+l = (1 − λ t 1 t+ l−1 

2

(decα )i .

2

 −1 1/2 ω Here, it is denoted that Ei  R˜1i /2 ν˜i + Ci ij= ˜ j and t i, j+1 Q t+l−1 Ft  j=t t+l, j+1 Q 1/2 ω ˜ j. The estimation error of the filter (28) is defined as x¯¯t+l  Eγ˜t ,...,γ˜t+l (xˇt +l |t +l−1 − xt+l ). It can be further written as,

1 −1 O˜ t+ l,t

 i=1

i, j+1 Q 1/2 ω˜ j ,

t+l, j+1 Q 1/2 ω˜ j .

(40)

lim E[x¯¯tT x¯¯t ] < lim E{x¯¯Trem(t,l ) x¯¯rem(t,l ) }(decα )mod (t,l )

t→+∞

mod (t−1,l )

j=t

˜ +λ

(39)

Then, it can be got that

+F

j=t

T t+l |t Tt [

(38)

The Eq. (32) can be written as

The state xt+l and measurement yi can be obtained by

yi = Ci i,t TtT (Tt xt ) + R˜1i /2 ν˜i + Ci

(37)

By replacing x = Tt x¯¯t into the Eq. (37), it can be got that,

2

i−1 

(35)

(36)

T ¯ ¯T ¯ ¯ ¯T ¯ E(x¯¯t+ l x¯t+l ) = Ex¯¯t [E (x¯t+l x¯t+l |x¯t )] ≤ decα E (x¯t x¯t ) + F ,



t+ l−1 

(34)

T ¯ ¯ ¯T ¯ E(x¯¯t+ l x¯t+l |x¯t ) ≤ decα x¯t x¯t + F

where

xt+l = t+l,t TtT (Tt xt ) +

(33)

i=t

2



T T ˇi i,t Ci y

i=t

yˇi =



Ti,t CiT Ei + FtT Ft .

By the definition of O˜ t+l,t , for any x, it can be got that,

≤ E[(xˇt |t −1 − xt )T (xˇt |t −1 − xt )]

T t+l,t Tt [

1 −1 TtT [(O˜ t+  0] l,t )

Ft ≤ F < +∞.

decα and decβ are same as the parameters in Definition 1.

where



×

Theorem 1. If Assumption 1 is satisfied and the pair (At , Ct )  is unidecβ obs

T

Ti,t CiT Ei

T T −1 ˜1 × Tt t+  0]Tt l |t t+l |t Tt [ (Ot+l,t )

It is easy to note that for the system (10) this assumption also means R˜t ≤ (r + η/βG2 )I. formly observable, there will exist a critical value λc = 1 −

 i=t

(25)

1 O˜ t+ l,t

(32)

where the second part of (32) is,

c ≤ Ct CtT  ≤ c ,

qI ≤ Q ≤ qI,

223

(31)

Section 3 puts forward the stochastic boundedness conditions for the PECMS. However, the proof of Theorem 1 only gives a conservative bound for E(Pˆt+1|t ) because the exact value of decβ and

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X. Liu et al. / Signal Processing 138 (2017) 220–232



obs is hard to acquire in view of the practical application. Therefore, in this section, a rational bound for E(Pˆt+|t ) is proposed for the system (10) with the scalar measurement, which means yt ∈ R. Because the dimension of yt becomes one in this case, it is therefore reasonable to represent Ct , R and  as ct , r and χ , respectively. By combining (13) and (15), the iteration of Pˆt |t −1 in the case

× 1−

λ˜ a2 c21 ρ (Pt )2 I 2 c2 ρ (Pt ) + r   2 ˜ λχ a2 c21 ρ (Pt )2 c2 ρ (Pt ) + r + ln 1 + I χ (c22 ρ (Pt ) + r )(c22 ρ (Pt ) + r )  λ˜ a2 c2 ρ (Pt )2 2 ≤ a ρ (Pt ) + q − 2 1 c2 ρ (Pt ) + r  2 ˜ λχ a2 c21 c2 ρ (Pt ) + r + ln 1 + I 2 χ c2 c22 2

≤ a

of scalar measurement becomes

Pˆt+1|t = At Pˆt |t −1 AtT + Q −

γ˜t At Pˆt |t −1 ctT ct Pˆt |t −1 AtT . ct Pˆt |t −1 ctT + r + χ /˜ιt

(43)

For convenience, it is denoted that Pt  E(Pˆt |t −1 ). Because the right hand side of (43) is a concave function of Pˆt |t −1 [21, Lemma 1(e)], by the Jensen’s inequality, it could be deduced by taking expectation on both side of (43) that,



Pt+1

γ˜t At Pt ctT ct Pt AtT ≤ Eγ˜t ,˜ιt +Q − ct Pt ctT + r + χ /˜ιt  T T ˜ιt ˜ At Pt ct ct Pt At E˜ι = At Pt AtT + Q − λ , t βt + ˜ιt ct Pt ctT + r At Pt AtT

where βt =

χ

ct Pt ctT +r

(44)

.

∞ It is well known that exp(θ )E1 (θ ) = 0 exp(−t )/(t + θ )dt,

∞ where E1 (θ ) = θ exp(−t )/t dt . Considering the distribution of ˜ιt defined by (9), it can be deduced that,



˜ Pt+1 ≤ At Pt AtT + Q − λ ˜ = At Pt AtT + Q − λ



At Pt ctT ct Pt AtT E˜ιt 1 − ct Pt ctT + r

βt



(45)

Using the inequality exp(θ )E1 (θ ) ≤ ln(1 + 1/θ ), it can be derived that,

≤

At Pt AtT



βt + βG2  1



T T ˜ At Pt ct ct Pt At 1 − βt ln 1 + +Q −λ ct Pt ctT + r



1

1 − βt ln 1 +



βt

 k1 (Pt ).

(46)

By [30, Lemma 3.1], it is satisfied that k1 (P ) < k1 (P˜ ) when P < P˜. Denote the spectral radius of Pt as ρ (Pt ), it can be got that Pt ≤ ρ (Pt )I because Pt is symmetric and positive. Then,

k1 (Pt ) ≤ k1 (ρ (Pt )I ) =

2

Pt At ctT ct AtT Pt ct ctT + r

(47)

Assumption 2. E(Pˆ1|0 ) > 0 and there exist the real numbers a, a, c1 , c 1 , c2 , c 2 , q > 0, such that:

a ≤ At  ≤ a,

Q  ≤ q, c1 ≤ ctT ct  ≤ c1 , c2 ≤ |ct ctt | ≤ c2 . Under Assumption 2, by the fact that xln(1 + it can be deduced that,

ρ (Pt )I + qI −

(50)

(51)

by the equations of (45) and (49) and the increasing property of k1 (P),  5. The stochastic boundedness of the estimation error Based on the prior boundedness of error covariance matrices, this section presents the main result of this paper that E(et |t −1 2 ) is bounded under appropriate off-line conditions. The analysis of the estimation error is of great importance because the previous stochastic boundedness of the PECMS cannot equivalently guarantee the stochastic boundedness of the estimation errors in MEKF. The results regarding the bound of E[Pˆt+1|t ] in previous Sections 3 and 4 also play the important roles in the stochastic boundedness of the estimation error.

pI ≤ E(Pˆt |t −1 ) ≤ pI.

It is obvious that the matrix function k1 (Pt ) needs the on-line results of filtering, i.e., At , ct . To acquire an off-line upper bound on Pt , some assumptions are further needed.

2

Proof. The theorem is proven by the mathematical induction. On one hand, by (45) and (49), Z1 = P1 and ξ1 = ρ (P1 ) result in E(Pˆ2|1 ) ≤ Z2 ≤ ξ2 I. On the other hand, if it is assumed that E(Pˆt |t −1 ) ≤ Zt ≤ ξt In , it will be got that,

Assumption 3. There exist the real numbers p and p such that

ρ( ) ρ (Pt )At AtT + Q − λ˜ ρ( )   ρ (Pt )ct ctT + r χ × 1− ln 1 + I. χ ρ (Pt )ct ctT + r

k1 (ρ (Pt )I ) ≤ a

(49)

Theorem 2. If the scalar measurement system satisfies Assumption 2, there will exist two upper bound sequences {Zt } = k1 (Zt−1 ) and {ξt I} = k(ξt−1 )I, which satisfies Z1 = E(Pˆ1|0 ) and ξ1 = ρ (E(Pˆ1|0 )), respectively. It is held that E(Pˆt |t −1 ) ≤ Zt ≤ ξt In , ∀t.

Zt+1 = k1 (Zt ) ≤ k1 (ξt I ) ≤ k(ξt )I ≤ ξt+1 I,

× 1 − βt exp(βt + βG2 )E1 (βt + βG2 ) .

˜ Pt+1 ≤ At Pt AtT + Q − λ

 k(ρ (Pt ))I.

and





ρ (Pt )I + qI −

E(Pˆt+1|t ) = k1 (E(Pˆt |t −1 )) ≤ k1 (Zt ) = Zt+1 ,

βt + ˜ιt

At Pt ctT ct Pt AtT ct Pt ctT + r

At Pt ctT ct Pt AtT ct Pt ctT + r

 ρ (Pt )ct ctT + r χ ln 1 + I χ ρ (Pt )ct ctT + r

λ˜ a2 c21 ρ (Pt )2 2 c2 ρ (Pt ) + r

(48) 1 x)

< 1 for x > 0,

(52)

Moreover, the residuals of the Taylor series (11) are bounded for any xt , xˆt |t −1 and xˆt |t ∈ Rn , i.e.,

φ (xt , xˆt |t ) ≤ φ et |t 2 , χ (xt+1 , xˆt+1|t ) ≤ χ et+1|t 2 ,

(53)

where  φ and  χ are the bounded positive real numbers, i.e., 0 <  φ ,  χ < ∞. Remark 2. The two parts of Assumption 3 are reasonable. On one hand, the first part regarding E(Pˆt |t −1 ) can be provided by Theorems 1 and 2. On the other hand, the rationality of the second part is guaranteed by the Taylor series theorem which shows that,

φ (xt , xˆt |t ) =

etT|t H f |x´t |t et |t ,

T χ (xt+1 , xˆt+1|t ) = et+1 |t Hh |x`t+1|t et+1|t ,

(54)

X. Liu et al. / Signal Processing 138 (2017) 220–232

where the H f |x´

is the Hessian matrix of the function f at x´t |t , which is a vector between xt and xˆt |t , and the Hh |x` is the Hest |t

t+1|t

sian matrix of the function h at x`t |t , which is also a vector between xt+1 and xˆt+1|t . If the Hessian matrix Hf and Hh are bounded for ∀x, the second part of Assumption 3 is therefore satisfied. In order to further analyze the boundedness of the estimation error, the standard result about the Banach fixed-point theorem is recalled as follows [40]. Lemma 1. (Fixed-point Theorem) [40] If g(x) is a continuous function with g(x) ∈ (a, b) and |g (x)| < 1 for all x ∈ (a, b), g(x) will have a unique fixed-point x∗ ∈ (a, b) such that x∗ = g(x∗ ). Furthermore, x∗ can be determined as follows: start with an arbitrary element x0 ∈ (a, b), and define a sequence {xt } by xt+1 = g(xt ), then xt → x∗ when t → +∞. Now, the main result of this paper can be stated as follows. Theorem 3. The MEKF in Section 2.3 is taken into consideration, which is used to estimate the state of the system (10) with both transmission failure and signal fluctuation in Section 2.1. If Assumptions 1 and 3 hold, and the constraint is met that 3/2

−α1 (1 )2 + α2 p

(1 )3 + α3 p2 (1 )4 + α4 < 0,

(55)

where



1 =

4α3

,

(56)

the estimation error et |t −1 will be stochastically bounded, i.e., E(et |t −1 2 ) < +∞, when the initial estimation error covariance R0 satisfies that,



2

R0  ≤

Secondly, when γ˜t = 1, the iteration of E(Vt+1 (et+1|t )) is

Eγ˜t =1 (Vt+1 ) −1 = E{etT|t −1 (At − At Kt Ct )T Pˆt+1 |t (At − At Kt Ct )et |t −1 −1 T ˆ−1 + mtT Pˆt+1 |t [2(At − At Kt Ct )et |t −1 + mt ] + st Pt+1|t st }.

−a +

a − 4φ (q − 1 )

−1 E[etT|t −1 (At − At Kt Ct )T Pˆt+1 |t (At − At Kt Ct )et |t −1 ]

= E[etT|t −1 [Pˆt |t −1 + (At − At Kt Ct )−T ×(At Kt R˜t KtT AtT + Q )(At − At Kt Ct )−1 ]−1 et |t −1 ] q ≤ (1 + )−1 E[Vt (et |t −1 )], (a + akc )2 p

2φ

.

(57)

˜ ,  , β G ,  χ ,  a, The parameters α 1 , α 2 , α 3 and α 4 are relative to λ φ c, q, r, η, a, c, q, p and p, which will be elaborated in the following proof.

pc . pc2 + r

k=

T −1 E[[2(At − At Kt Ct )et |t −1 + mt ] Pˆt+1 |t mt ]

+

(φ + χ ak )2 p

The boundedness of E(et |t −1 2 ) is equivalent to the boundedness of E(Vt+1 (et+1|t )). Next, it will be shown that E(Vt (et |t −1 )) is bounded when t → +∞. Firstly, when γ˜t = 0, the iteration of E(Vt+1 (et+1|t )) is

Eγ˜t =0 (Vt+1 (et+1|t ))

p (φ + χ ak )2 2 E [Vt (et |t −1 )] p 2

≤

+

2p

p

q

−1 E[stT Pˆt+1 |t st ] ≤

E(Vt (et |t −1 )) +

2

≤

1+

q

−1

2

a p 3/2

+

2aφ p p

t

2 2

q a k = + [r + ηE1 (βG2 )]. p p

(64)

By combining (61), (63) and (64), when γ˜t = 1, the iteration could be finalized that,

Eγ˜t =1 (Vt+1 (et+1|t )) ≤ (1 − [1 − (1 +

q

)−1 ] )E(Vt (et |t −1 )) (a + kac )2 p 3/2 2 p (1 + akc )(φ + χ ak ) 3/2 + E (Vt (et |t −1 )) +

φ2 p

+ E(et |t −1 4 )

p (φ + χ ak )2 2 E (Vt (et |t −1 )) p

E(Vt (et |t −1 )) + E3/2 (Vt (et |t −1 )) +

φ2 p2 p q . p

q a k + (r + ηE1 (βG2 )) p p

(65)

Then, by taking the expectation on γ˜t of the iteration, it can be got that,

2aφ −1 + E(et |t −1 3 ) + E(ωtT Pˆt+1 |t ωt ) p



2 2

q a k 1 + [r + ηEϑ˜t ( )] p p ϑ˜ 2

2 2

−1

a p

(63)

p

−1 + ωtT Pˆt+1 |t ωt }

1+

E3/2 [Vt (et |t −1 )].

2

T −1 + [2At et |t −1 + φ (xt , xˆt |t −1 )] Pˆt+1 |t (φ (xt , xˆt |t −1 ))

≤

(1 + akc )(φ + χ ak )

The third term of the right-hand side in (60) is

= E{etT|t −1 (Pˆt |t −1 + At−T QAt−1 )−1 et |t −1



Eτ1 ,...,τt [et |t −1 4 ]

2(1 + akc )(φ + χ ak ) E[et |t −1 3 ] p

that,

(58)

(62)

The second term of the right-hand side in (60) can be further derived as

T ˆ−1 Proof. Define Vt+1 (et+1|t ) = et+1 |t Pt+1|t et+1|t . From (52), it is got

1 1 e 2 ≤ Vt+1 (et+1|t ) ≤ et+1|t 2 . p t+1|t p

(61)

where

3/2

4

(60)

The first term of the right-hand side in (60) can be further formulated as

≤ 9α22 /4 + 8α1 α3 − 3α2 /2

225

E(Vt+1 (et+1|t )) ˜ )Eγ˜ =0 (Vt+1 (et+1|t )) ˜ Eγ˜ =1 (Vt+1 (et+1|t )) + (1 − λ =λ t t

E2 (Vt (et |t −1 ))

≤ (1 − α1 )E(Vt (et |t −1 )) + α2 E3/2 (Vt (et |t −1 )) (59)

+α3 E2 (Vt (et |t −1 )) + α4  g(E[Vt (et |t −1 )] ),

(66)

226

X. Liu et al. / Signal Processing 138 (2017) 220–232

where

  α1 = λ˜ 1 − 1 +

q

(a + akc )2 p

˜ )[1 − (1 + + (1 − λ

α2 = λ˜ [

3/2

2p

q 2

a p

)−1 ],

(1 + akc )(φ + χ ak )

 ˜) + (1 − λ

Remark 3. The corresponding theorem of Reif et al. [41] and Kluge et al. [26] could only be considered as a method to verify the stability of the filter during the state estimation process because it needs the on-line result of et |t −1 , which is hard to acquire in practice. Although Theorem 3 needs the stricter constraint compared with Reif et al. [41] and Kluge et al. [26], it can effectively ensure the estimation error of MEKF bounded in the mean-square sense by off-line conditions, which is meaningful as the performance of MEKF can be guaranteed by only checking the system and channel parameters.

−1

p 3/2

2aφ p p

]

,

 2  2 2  p p (φ + χ ak )2 ˜) φ α3 = λ˜ + (1 − λ , p

p

  2 2 q a k q 2 ˜ ˜ α4 = λ + (r + ηE1 (βG )) + (1 − λ ) . p

p

(67)

p

It is obvious to derive that 0 < α 1 < 1 and α 2 , α 3 , α 4 > 0. Considering the inequality (66), a similar progression St is developed as follows, where S1 = E(V1 (e1|0 )) > 0, and

St+1 = g(St )  St + g1 (St ).

(68)

Next, by using Lemma 1, the existence of a positive bounded S∗ will be shown such that lim St → St∗ , if both (55) and (57) hold. t→+∞

Firstly, it is needed to demonstrate that g (x) < 1. For this propose, the derivative function of g1 (x) can be analyzed as follows

g1 (x ) = 2α3 x + 3/2α2 x1/2 − α1 .

(69)

On one hand, because α 2 , α 3 > 0, g1 (x ) is continuously and monotonically increasing for x > 0, which means g1 (0 ) < g1 (x ) < g1 (1 ) for x ∈ (0,  1 ). On the other hand, it is obvious that g1 (0 ) = −α1 < 0 and lim g (x ) = +∞ > 0. Because g (x ) is continuous, there x→+∞ 1

1

must exist an  1 > 0 so that g1 (1 ) = 0. In the case that g1 (1 ) = 0 and g1 (0 ) = −α , it can be thus deduced that −1 < −α < g1 (x ) < 0, i.e., 0 < g (x) < 1 for x ∈ (0,  1 ). Secondly, it will be shown that g(x) ∈ (0,  1 ) for x ∈ (0,  1 ) under the condition (55). By considering g (x ) = g1 (x ) + 1 > 0 for x ∈ (0,  1 ), it can be got that g(0) < g(x) < g( 1 ) for x ∈ (0,  1 ). Moreover, it is noted that the left side of (55) is exactly identical to g1 ( 1 ). Therefore, it can be deduced that g1 ( 1 ) < 0, which also means g( 1 ) <  1 . Then, due to the fact that g(0 ) = α4 > 0, it can be got that 0 < g(x) <  1 for x ∈ (0,  1 ). To conclude, g(x) therefore meets the conditions of Lemma 1. Now, the initial element of the sequence {E(Vt (et |t −1 ))} is taken into consideration. By (15), there holds that

e1|0 ≤ ae0 + ω0 + x˜0 2 .

(70)

Then, it can be equivalent to,

E[V1 (e1|0 )] ≤ a

2

R0  + q + φ R0 2 .

(71)

If the initial condition (57) is guaranteed, it can be obtained that S1 = E(V (e1|0 )) ∈ (0, 1 ). Finally, it can be got that lim St → St∗ , where S∗ ∈ (0,  1 ) by Lemma 1. Next, let us turn to consider E(V1 (e1|0 )) = S1 , it is easy to get that,

t→+∞

E(Vt (et |t −1 )).

E(V2 (e2|1 )) ≤ g(E(V1 (e1|0 ))) = g(S1 ) = S2 .

Because

(72)

Moreover, because g(x) is monotonically increasing, E(Vt (et |t −1 )) ≤ St implies that,

E(Vt+1 (et+1|t )) ≤ g(E(Vt (et |t −1 ))) ≤ g(St ) = St+1

(73)

By the inductive method, it can be proven that E(Vt (et |t −1 )) ≤ St for ∀t. Therefore, it can be inevitably concluded that E(Vt (et |t −1 )) is bounded as t → +∞. Therefore, it also equivalently holds that E(et |t −1 2 ) is bounded as t → +∞ by (58). 

Remark 4. The proof of Theorem 3 shows that it is important to introduce the drop strategy. The absence of the drop strategy equals that βG = 0, which will make α4 = +∞ in (67) because E1 (0 ) = +∞. β G must be carefully chosen because it should be large enough to make α 4 as small as possible to satisfy the condition (55) in Theorem 3. Meanwhile, it should be as small as pos˜ defined by (8) bigger than the critical rate λc in sible to ensure λ Theorem 1. 6. Numerical example and verification In this section, the proposed theorems will be verified to highlight various parameters’ influence on the stability, such as the arrival rate λ and the initial estimation error R0 , the existent ˜ c , the effectiveness of the upper bound of the critical value λ for the PECMS, the strategy of the choice of β G and the impact of the channel additive noise . For this purposes, a nonlinear discrete-time system, which is widely used in nonlinear filtering problems [36,41,42], is adopted. The system is a model for the initial alignment of SINS with large misalignment angles, and is presented as follows:



x1,t+1 x2,t+1





=

yt =



f1 (x1,t , x2,t ) + f2 (x1,t , x2,t )



ω1,t , ω2,t

x1,t + νt ,

(74)

where

f 1 ( x1 , x2 ) = x1 + τ x2 , f2 (x1 , x2 ) = x2 + τ (−x1 + x2 (x21 + x22 − 1 )),

(75)

and τ = 0.003.[ω1,t , ω2,t ] and ν t are zero mean Gaussian noises with covariance Q = diag[0.0 032 , 0.0 032 ] and R = 0.012 , respectively. Except in Section 6.3, the initial noise covariance is chosen as R0 = diag[0.52 , 0.52 ] in all other simulation of Section 6. The initial states of the system are chosen as [x1, 0 , x2, 0 ] = [0.8, 0.2], and the initial estimation [xˆ1,0 , xˆ2,0 ] is generated from N ([x1,0 , x2,0 ], R0 ). The observation yt is transferred through the fading channel described in Section 2.1 with both transmission failure and signal fluctuation, which will be described in Section 6.1. T

6.1. Demonstration of channel fading and drop strategy The fading channel model is described as the Eq. (5), where γ t reflects the transmission failure, ϑt is the signal fluctuation, and ηt is the channel additive noise. In this simulation, γ t is chosen as a Bernoulli process with the arrival rate p(γt = 1 ) = λ, which will be chosen separately in each simulation. ϑt follows the Rayleigh fading, which means ϑt2 ∼  exp(−ϑt2 ). Also, the Rayleigh parameter  will be chosen elaborately in each simulation. At last, ηt is a zero mean Gaussian noise with covariance £ which will also be altered in every simulation. As discussed in Section 2.2, the drop strategy also plays an important role in the estimation process as a part of the MEKF. To illustrate this, a simulation for the fading channel including the fluctuation ϑt and intermittent γ t of one sample path is conducted with the channel additive

X. Liu et al. / Signal Processing 138 (2017) 220–232

227

Table 1 Initial value and arrival rate for the simulation of the MEKF for the system (74).

R0

λ

Normal

Low arrival rate

Large initial noise

0.52 0.8

0.52 0.2

2.52 0.8

6.2. Performance comparison of filter considering channel condition

Fig. 1. Demonstration of the influence of channel fading and drop strategy when λ = 0.8,  = 5.5 and βG2 = 0.1.

noise  = 0.12 , the drop rate λ = 0.8, the Rayleigh fading parameter  = 5.5 and the drop strategy gate βG2 = 0.1. The simulation results are shown in Fig. 1. Three cases can be categorized during the process when the observations are transformed through the fading channel. The first case, marked in red, is that the filter does not get the observation because the transmission failure happens, which means γt = 0. In another case, marked in green, the filter gets the observation and drops it by the drop strategy when ϑt < β G . In the remaining case, marked in blue, the observation is got by the filter and is utilized for the estimation process. Algorithm 1 is estab-

Algorithm 1 Simulation process of MEKF under fading channel. Initialization: Generate xˆ0|0 ∼ N (x0 , R0 ). Loop Process: while t ≥ 1 do xˆt |t −1 = f (xˆt −1|t −1 ) At−1 = ∂ f (x )/∂ x|x=xˆ t −1|t −1

Pˆt |t −1 = At−1 Pˆt −1|t −1 At−1 + Q Generate ι ∼  exp(−ι ) √ ϑ= ι Generate η ∼ N (0, ) zt = yt + η/ϑ if ι ≥ βG2 then Generate ς from uniformly distributed on (0, 1 ) if ς < λ then γ˜ = 1 else γ˜ = 0 end if else γ˜ = 0 end if if γ˜ == 1 then Ct = ∂ h(x )/∂ x|x=xˆ t |t −1

Kt = Pˆt |t −1CtT (Ct Pˆt |t −1CtT + R + /ϑ 2 )−1 xˆt |t = xˆt |t −1 + Kt (zt − h(xˆt |t −1 )) Pˆt |t = Pˆt |t −1C T (Ct Pˆt |t −1C T + R + /ϑ 2 )−1Ct Pˆt |t −1 t

t

else xˆt |t = xˆt |t −1 Pˆt |t = Pˆt |t −1 end if end while

lished to detail the MEKF filtering theory and simulation process.

As the observation function of the system (74) is linear, the filter proposed in [38] can be utilized to estimate the states of the system (74) under both transmission failure and signal fluctuation. For convenience, the one step prediction of filter in [38] is defined as x˜t |t −1 and the one step prediction estimation error as e˜t |t −1 . In this part for comparison, the other parameters are chosen as  = 0.022 ,  = 5.5 and βG2 = 0.05. In order to clearly demonstrate the estimation accuracy comparison, a high arrival rate is chosen as λ = 0.95. The filtering results are shown in Fig. 2. Fig. 2(a) shows that the estimation error of MEKF is much smaller than the filter in [38], which is also confirmed in Fig. 2(b) that MEKF can track the state x2 quickly while it takes much more time for the filter in [38] to track the state. Besides, the other criteria to evaluate the filter under fading channel is the crucial arrival rate. For this purpose, we reduce λ progressively to produce a poor communication condition. The filtering results are shown in Fig. 3 when λ = 0.82. Under that channel condition, the filter in [38] becomes divergent as shown in Fig. 3(a). At the same time, the MEKF still tightly tracks the system states, and provides the proper estimation accuracy. 6.3. Effect on stability of λ and R0 Because the system has the uniformly observable property [41, Section 5], the boundedness of the PECMS can be guaranteed if ˜ is large enough by Theorem 1. Then, the filtering arrival rate λ by using Theorem 3 and combining the boundedness of other parameters (i.e., At , Ct , Q, R and ) by the simulation methods in [41, Section5], the estimate error will be bounded if the initial error covariance R0 is small enough, the drop strategy gate β G is appropriate and the channel additive noise  is small enough. ˜ ) are In this part, R0 and λ (which is directly proportional to λ taken into consideration to verify the stability of the MEKF. The ˜ and the influence of β G and  existence of the critical value λ will be further shown in the next part. In order to verify λ and R0 , three cases are specifically demonstrated, including the normal case with the small initial error and high arrival rate, the low arrival rate case with the small initial error and the large initial error case with high arrival rate. The channel parameters, except λ, are chosen the same as the previous simulation of the fading channel. The parameters λ and R0 are detailed in Table 1. The other parameters in this simulation are chosen as  = 0.12 ,  = 5.5 and βG2 = 0.1. The simulation results are shown in from Figs. 4 to 6, where both the estimation error et |t −1  and the one step prediction xˆ2,t are illustrated for each case. It can be verified from the simulation that a low arrival rate inevitably makes the estimation errors unbounded because the corresponding low arrival rate will cause the PECMS unbounded by Theorem 1, which therefore results in the unbounded estimation error. At the same time, the large initial error also results in the divergence of the estimation error by Theorem 3. 6.4. Verification on the existence of λc Fig. 4 shows the stable filtering results. On the contrary, Figs. 5 and 6 show the divergent filtering results. Theorem 1 discusses

228

X. Liu et al. / Signal Processing 138 (2017) 220–232

Fig. 2. Comparison of MEKF with filter in [38] when λ = 0.95.

Fig. 3. Comparison of MEKF with filter in [38] when λ = 0.82.

Fig. 4. Simulation results of the MEKF for the system (74) when λ = 0.8 and R0 = 0.12 .

that there exists a critical value of the arrival rate λc such that the filter is stable if the filtering arrival rate exceeds the critical value ˜ > λc ). As discussed in Section 2.1, λ ˜ satisfies the Eq. (8), (i.e., λ ˜ is related to λ,  and β G . Therefore, two simulations are where λ accordingly designed to verify the existence of λc . In the first simulation, the parameters R0 , β G ,  are set as R0 = 0.52 , βG2 = 0.1 and  = 0.12 and five values of the fluctuation parameter  are selectively chosen as in Table 2. For each  , the arrival rate λ is accordingly set one by one from 0 to 1 with the interval of 0.02 to perform the filtering. By 500 Monte Carlo simulations, the result is shown in Fig. 7. Average estimation error 1000 of et |t −1  over 10 0 0 points (i.e., 10100 t=1 et |t −1 ) is plotted for each λ only when the filter is stable under the corresponding settings of λ and  . Note that the average estimation error is different on each  curve for the same λ because the mean of

Table 2 The choices of  and simulation results of λsimu for λc .



exp(−βG2 )

λsimu λsimu · exp(−βG2 )

1

2

3

4

5

0.0100 0.9900 0.18 0.1782

1.1541 0.8910 0.20 0.1782

2.1072 0.8100 0.22 0.1782

2.9773 0.7425 0.24 0.1782

3.7778 0.6854 0.26 0.1782

ιt = ϑt2 is 1/ , which makes the covariance matrix of equivalent observation noise R˜t different. Therefore, the smallest λ of the stable filtering under certain  can be found and is denoted as λsimu , which is recorded in the third rows of Table 2. Then, λsimu · exp(−βG2 ) could be regarded as the simulation results of the λc , which is also shown in the last rows of Table 2.

X. Liu et al. / Signal Processing 138 (2017) 220–232

229

Fig. 5. Simulation results of the MEKF for the system (74) when λ = 0.2 and R0 = 0.12 .

Fig. 6. Simulation results of the MEKF for the system (74) when λ = 0.8 and R0 = 2.52 .

Table 3 The choices of β G and simulation results of λsimu for λc .

βG

exp(−βG2 )

λsimu λsimu · exp(−βG2 )

β G1

β G2

β G3

β G4

β G5

0.10 0 0 0.9900 0.18 0.1782

0.3397 0.8910 0.2 0.1782

0.4590 0.8100 0.22 0.1782

0.5456 0.7425 0.24 0.1782

0.6147 0.6854 0.26 0.1782

Fig. 7. Simulation results of λsimu for different  when βG2 = 0.1 and  = 0.12 .

Similarly, the same parameters R0 = 0.52 ,  = 1 and  = 0.12 are used in the second simulation, and five different values of β G are chosen, which is declared in Table 3. For each β G , λ is selectively chosen in the same way as the first simulation, and the corresponding average estimation errors are plotted in Fig. 8 as well. Because λsimu · exp(−βG2 ) stands for the simulation result of λc , the existence of λc can be thus verified by the last row of Tables 2 and 3. Note that the average estimation error is almost

Fig. 8. Simulation results of λsimu for different β G when  = 1 and  = 0.12 .

230

X. Liu et al. / Signal Processing 138 (2017) 220–232

Fig. 9. PECMS and its bounds of the MEKF for the system (74).

identical for the same λ even when β G is different because R˜t is same although β G is different. To conclude, λc definitely exists and has no relationship with the energy level of the channel noise. It is only dependent on the model of signal fluctuation, drop strategy gate and the drop rate, which verifies the Eq. (8). 6.5. Verification on the upper bound of PECMS By Theorem 1, the PECMS is stochastic bounded under the uniform observability condition. The system (74), whose dimension of the observation is one, satisfies the scalar observation condition of Theorem 2. The simulation in this part is to demonstrate the performance of the upper bound of the PECMS proposed in Theorem 2. Note that it is meaningless to analyze the bounds of PECMS of a divergent filter so that only the normal stable case is under consideration. The effect of  is compared to demonstrate the explicit bounds of the system, which is plotted in Fig. 9. It is shown that the upper bounds ξ t is positively correlated with , as shown in the Eq. (49). 6.6. Effect of β G on performance of filtering

Fig. 10. Simulation results for the network.

+ (I − Kt )Qx (I − Kt )T /ϑ˜t2 + Kt Rx KtT /ϑ˜t2 .

(77)

By comparing the Eqs. (76) with (77), it is clear that there exists a value ϑc that Pˆt+1,keep > Pˆt+1,drop when ϑ˜t < ϑc . Thus, when ϑ˜t < ϑc , we would rather drop the observation by setting γ˜ = 0 k

Under the condition that the PECMS is stochastically bounded, the appropriate β G ,  and R0 are of critical importance for the sufficient conditions for the stochastic boundedness of the estimate error by the equations of (55), (67) and (57) in Theorem 3. The influence of R0 has been shown in the previous part. Therefore, the relationship between the β G and the average estimate error is shown in this part. The parameters are chosen as λ = 0.8 and  = 1. Also, two different additive noise  = 0.052 and  = 0.0752 is selected for comparison. The simulation results are shown in Fig. 10. It is obvious that there exists an optimal β G minimizing the average estimate error. Therefore, β G should be appropriately designed, which can be neither too large nor too small. It is similar with the result in [38]. Remark 5. To demonstrate the selection strategy of β G , φ (xt , xˆt |t ) and χ (xt , xˆt |t ) in the Eqs. (16) and (17) are approximated by two zero mean noises with covariance Qx and Rx , respectively. For convenience, the assumption is made that the two noises are independent with ωt , ν t , ϑt and each other. Then, the covariance of estimation by dropping the observation is that,

than utilize yk to perform the measurement update step of filtering. In this case, we should set β G = ϑc . By the Eq. (77), the β G has the positive correlation with Qx and Rx , which means that β G should be chosen according to the nonlinearity of system and observation functions. Especially, when the nonlinearity of observation functions becomes notable, a suitable drop strategy β G should be chosen selectively, where Kt Rx KtT /ϑ˜t2 will become a nonnegligible large value with small ϑ˜t . However, the selection of β G by precisely calculating the Eqs. (76) and (77) always needs a mass of computation since it is hard to calculate Qx and Rx . A feasible replacement to select β G is the practical method through simulation or experiment. Remark 6. For the system (74), it should be noticed that the observation function is linear. By the Eq. (77), the influence of ϑ˜t on Pˆt+1,keep is slight. In Fig. 10, the simulation result confirms the analysis, where the estimation error are almost the same even β G is small. 6.7. Effect of additive noise with bounded PECMS

Otherwise, the covariance of estimation by keeping the observation is that,

From the Eqs. (55) and (67) in Theorem 3, it can be concluded that an improper  will make the estimate error divergent even when the E[Pˆt+1|t ] is bounded. In order to demonstrate that, a system with strong non-linearity is introduced here as:

Pˆt+1,keep = Pt+1|t − Pˆt+1|t CtT (Ct Pˆt+1|t CtT + R˜t )−1Ct Pˆt+1|t

xt+1 = 10 sin(xt ) + xt + ωt ,

Pˆt+1,drop = At Pˆt |t AtT + Q + Qx = Pˆt+1|t + Qx .

(76)

X. Liu et al. / Signal Processing 138 (2017) 220–232

231

Fig. 11. Estimate error of the MEKF for the system (78).

Fig. 12. PECMS of the MEKF for the system (78).

Fig. 13. State and estimate of the MEKF for the system (78).

yt = 10 cos(xt ) + 11xt + νt ,

(78)

where ωt and ν t are the white Gaussian noise with the covariance matrix Qk = 0.0032 and Rk = 0.012 , respectively. The initial state is x0 = 10 and the noise of the initial estimator is R0 = 0.012 . The observed data is transformed through the channel modeled as in Section 2.1 with λ = 0.95,  = 0.5 and βG = 0.01. The channel additive noise covariance matrix  is chosen to verify the theorem. Two cases are included into consideration. The first case stands for the small channel noise with  = 0.0012 , and the other case represents the large channel noise with  = 0.12 . The simulation

results are shown in Figs. 11–13, illustrating that the estimate error can be extraordinarily large even when Pˆt+1|t is still small, i.e. bounded. Therefore, the channel additive noise is another critical factor for the stochastic stability. 7. Conclusion This paper provides the off-line sufficient conditions on the mean-square boundedness of estimate error of the MEKF over the fading channel with both transmission failure and signal fluctua-

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tion. The critical filtering arrival rate is first proved to guarantee the stochastic boundedness of the PECMS, which only depends on off-line conditions. Based on the proposed upper bound of PECMS, the boundedness of estimation error is further addressed even when the PECMS is bounded and its corresponding sufficient conditions are also analytically provided. The influence of various parameters in proposed theorems, such as the initial error, channel additive noise and drop strategy gate and so on, is discussed in depth on the estimation performance and stability, which is also verified by different simulations. The main contribution of this work is that one can analyze the stochastic stability of a wireless estimation system using the filtering structure proposed in this paper. Our research interest will focus on analyzing the effect of transfer delay, quantization error on nonlinear filter with transmission failure and signal fluctuation in future work, which includes practical designs. Other triggering strategies of nonlinear filter will also be taken into consideration to reduce the communication relief. Acknowledgment This work was supported by National Natural Science Foundation of China under Grant Nos. 51407011, 11372034 and 11572035. References [1] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, Wireless sensor networks: a survey, Comput. Netw. 38 (4) (2002) 393–422. [2] A. Ribeiro, G.B. Giannakis, Bandwidth-constrained distributed estimation for wireless sensor networks-part i: gaussian case, IEEE Trans. Signal Process. 54 (3) (2006) 1131–1143. [3] J. Liu, S. Laghrouche, M. Harmouche, M. Wack, Adaptive-gain second-order sliding mode observer design for switching power converters, Control Eng. Pract. 30 (2014) 124–131. [4] S. Laghrouche, J. Liu, F.S. Ahmed, M. Harmouche, M. Wack, Adaptive second-order sliding mode observer-based fault reconstruction for pem fuel cell air-feed system, IEEE Trans. Contr. Syst. Technol. 23 (3) (2015) 1098–1109. [5] N. Liu, X. Lyu, Y. Zhu, J. Fei, Active disturbance rejection control for current compensation of active power filter, Int. J. Innov. Comput. Inf. Control 12 (2016a) 407–418. [6] J. Liu, W. Luo, X. Yang, L. Wu, Robust model-based fault diagnosis for pem fuel cell air-feed system, IEEE Trans. Ind. Electron. 63 (5) (2016b) 3261–3270. [7] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of markovian jump singular systems, IEEE Trans. Autom. Control 55 (5) (2010) 1213–1219. [8] L. Wu, W.X. Zheng, Reduced-order H2 filtering for discrete linear repetitive processes, Signal Process. 91 (7) (2011) 1636–1644. [9] N. Azman, S. Saat, S. Nguang, Nonlinear filter design for a class of polynomial discrete-time systems, Int. J. Innov. Comput. Inf. Control 11 (2015) 1011–1019. [10] P. Shi, X. Su, F. Li, Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation, IEEE Trans. Autom. Control 61 (6) (2016) 1694–1699. [11] X. Liu, Y. Yu, Z. Li, H.H. Iu, Polytopic H∞ filter design and relaxation for nonlinear systems via tensor product technique, Signal Process. 127 (2016) 191–205. [12] X. Su, P. Shi, L. Wu, Y.-D. Song, Fault detection filtering for nonlinear switched stochastic systems, IEEE Trans. Autom. Control 61 (5) (2016) 1310–1315. [13] J.P. Hespanha, P. Naghshtabrizi, Y. Xu, A survey of recent results in networked control systems, Proc. IEEE 95 (1) (2007) 138. [14] P. Shi, M. Mahmoud, S.K. Nguang, A. Ismail, Robust filtering for jumping systems with mode-dependent delays, Signal Process. 86 (1) (2006) 140–152. [15] L. Wu, Z. Wang, Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay, Signal Process. 89 (9) (2009) 1739–1753.

[16] X. Su, P. Shi, L. Wu, S.K. Nguang, Induced l2 filtering of fuzzy stochastic systems with time-varying delays, IEEE Trans. Cybern. 43 (4) (2013) 1251–1264. [17] X. Su, P. Shi, L. Wu, M.V. Basin, Reliable filtering with strict dissipativity for ts fuzzy time-delay systems, IEEE Trans. Cybern. 44 (12) (2014) 2470–2483. [18] R. Caballero-Águila, A. Hermoso-Carazo, J. Linares-Pérez, Fusion estimation using measured outputs with random parameter matrices subject to random delays and packet dropouts, Signal Process. 127 (2016) 12–23. [19] D.E. Quevedo, A. Ahlén, A.S. Leong, S. Dey, On kalman filtering over fading wireless channels with controlled transmission powers, Automatica 48 (7) (2012) 1306–1316. [20] D.E. Quevedo, A. Ahlen, K.H. Johansson, State estimation over sensor networks with correlated wireless fading channels, IEEE Trans. Autom. Control 58 (3) (2013) 581–593. [21] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, S.S. Sastry, et al., Kalman filtering with intermittent observations, IEEE Trans. Autom. Control 49 (9) (2004) 1453–1464. [22] M. Huang, S. Dey, Stability of kalman filtering with markovian packet losses, Automatica 43 (4) (2007) 598–607. [23] L. Xie, L. Xie, Stability of a random riccati equation with markovian binary switching, IEEE Trans. Autom. Control 53 (7) (2008) 1759–1764. [24] S. Kar, B. Sinopoli, J.M. Moura, Kalman filtering with intermittent observations: weak convergence to a stationary distribution, IEEE Trans. Autom. Control 57 (2) (2012) 405–420. [25] E.R. Rohr, D. Marelli, M. Fu, Kalman filtering with intermittent observations: on the boundedness of the expected error covariance, IEEE Trans. Autom. Control 59 (10) (2014) 2724–2738. [26] S. Kluge, K. Reif, M. Brokate, Stochastic stability of the extended kalman filter with intermittent observations, IEEE Trans. Autom. Control 55 (2) (2010) 514–518. [27] L. Li, Y. Xia, Stochastic stability of the unscented kalman filter with intermittent observations, Automatica 48 (5) (2012) 978–981. [28] L. Li, Y. Xia, Unscented kalman filter over unreliable communication networks with markovian packet dropouts, IEEE Trans. Autom. Control 58 (12) (2013) 3224–3230. [29] X. Su, L. Wu, P. Shi, Sensor networks with random link failures: distributed filtering for t–s fuzzy systems, IEEE Trans. Ind. Inform. 9 (3) (2013) 1739–1750. [30] S. Dey, A.S. Leong, J.S. Evans, Kalman filtering with faded measurements, Automatica 45 (10) (2009) 2223–2233. [31] D.E. Quevedo, A. Ahlén, et al., Energy efficient state estimation with wireless sensors through the use of predictive power control and coding, IEEE Trans. Signal Process. 58 (9) (2010) 4811–4823. [32] S. Ghandour-Haidar, L. Ros, J.-M. Brossier, On the use of first-order autoregressive modeling for rayleigh flat fading channel estimation with kalman filter, Signal Process. 92 (2) (2012) 601–606. [33] R. Parseh, K. Kansanen, On estimation error outage for scalar gauss–markov signals sent over fading channels, IEEE Trans. Signal Process. 62 (23) (2014) 6225–6234. [34] R. Parseh, K. Kansanen, Diversity effects in kalman filtering over rayleigh fading channels, IEEE Trans. Signal Process. 63 (23) (2015) 6329–6342. [35] N. Xiao, L. Xie, L. Qiu, Feedback stabilization of discrete-time networked systems over fading channels, IEEE Trans. Autom. Control 57 (9) (2012) 2176–2189. [36] L. Li, Y. Xia, Ukf-based nonlinear filtering over sensor networks with wireless fading channel, Inf. Sci. 316 (2015) 132–147. [37] A. Goldsmith, Wireless Ccommunications, Cambridge university press, 2005. [38] Y. Mostofi, R.M. Murray, To drop or not to drop: design principles for kalman filtering over wireless fading channels, IEEE Trans. Autom. Control 54 (2) (2009) 376–381. [39] B. Anderson, J.B. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Control Optim. 19 (1) (1981) 20–32. [40] V. Istrc, et al., Fixed Point Theory: An Introduction, 1, Springer, 1981. [41] K. Reif, S. Günther, E. Yaz, R. Unbehauen, Stochastic stability of the discrete– time extended kalman filter, IEEE Trans. Autom. Control 44 (4) (1999) 714–728. [42] G. Wang, J. Chen, J. Sun, Stochastic stability of extended filtering for non-linear systems with measurement packet losses, IET Control Theory Appl. 7 (17) (2013) 2048–2055.