Event-triggered state estimation for networked systems with correlated noises and packet losses

ISA Transactions xxx (xxxx) xxx

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Research article

Event-triggered state estimation for networked systems with correlated noises and packet losses ∗

Cui Zhu a , , Zhong Su b , Yuanqing Xia c , Li Li d , Juan Dai b a

School of Information and Communication Engineering, Beijing Information Science & Technology University, Beijing 100101, China Beijing Key Laboratory of High Dynamic Navigation Technology, Beijing Information Science & Technology University, Beijing 100101, China c School of Automation, Beijing Institute of Technology, Beijing 100081, China d School of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China b

article

info

Article history: Received 29 September 2018 Received in revised form 29 November 2019 Accepted 29 November 2019 Available online xxxx Keywords: Networked system Event-triggered communication State estimation Correlated noises Packet losses

a b s t r a c t The event-triggered state estimation for the systems suffering from correlated noises and packet losses is considered. A communication mechanism that determines the measurements to be sent or not depending on a specific event-triggered condition is presented to reduce the additional data transmissions. Then a novel event-triggered state estimator related to the trigger threshold and correlation coefficient is proposed. An expected trade-off between the rate of transmission and the estimator performance can be obtained through adjusting the threshold properly, and the influence of noise correlation and packet losses is weakened effectively. The estimator performance is evaluated and certain boundedness conditions for the covariance expectation are obtained. Finally, a target tracking system is supplied to support the relevant results. © 2019 Published by Elsevier Ltd on behalf of ISA.

1. Introduction State estimation for the networked systems that is widely used in various fields gains increasing attention of researchers. With the extensive application of network technologies and communication technologies, more and more devices including but not limited to sensors are connected to the network, and this has led to an explosion of data transmissions. How to design an applicable communication mechanism to transfer the data to the estimator effectively with the limited bandwidth and constrained energy is a problem to be solved urgently. Therefore, the event-triggered data transmission mechanism that determines the measurements to be sent or not has gradually turned into a research hotspot [1]. In [2], the event-triggered sampling for the first-order systems has been proved to perform better than the periodic one. A discrete-time type of event detector is employed to monitor the event trigger condition periodically in [3]. In [4] and [5], the data has been transmitted if the difference that is between the observations sent currently and the last-sent one exceeds a constant threshold. In [6] and [7], the innovation-based triggering has been implemented, where the difference that is between the current observations and its prediction instead of the last-sent one is used as the measure. In [8], a variance-based data ∗ Corresponding author. E-mail addresses: [email protected] (C. Zhu), [email protected] (Z. Su), [email protected] (Y. Xia), [email protected] (L. Li), [email protected] (J. Dai).

transmission mechanism has been proposed, where the measurement is sent only when the covariance is greater than a specific value. In [9], a novel scheduling mechanism has been introduced, in which the normalized innovation vector has been treated as a trigger condition. Based on that, a novel event-triggered filter is proposed. However, the Gaussian distribution assumption is not necessarily satisfied, which severely constrains the application of the proposed estimator. A further work eliminating the above approximations is studied in [10], where two different eventtriggered scheduling mechanisms that can be applied to openand closed-loop frameworks are proposed. All the results above have been achieved under the condition of uncorrelated noises. However, the noise correlation exists in many practical applications which will impact the performance of the filter severely. In this case, state estimation with correlated noises is widely taken into account in networked systems [11– 20]. A recursive filtering method has been proposed in [11], where the noises cross-correlation as well as auto-correlation exist. Feng et al. [12] have designed an optimal recursive Kalmantype filter dealing with the auto-correlated noises. A similar problem has been considered in [13], where there exist multiple packet losses in addition to noise auto-correlation and crosscorrelation. In [14], the centralized and distributed filters based on innovation for the systems with multiple sensors and lost observations are derived. Behbahani et al. [15] analyze the decentralized estimation under correlated noises, and point out that the system performance deteriorates if the noises at the fusion center are correlated. An Optimal sequential filter and a distributed filter

https://doi.org/10.1016/j.isatra.2019.11.038 0019-0578/© 2019 Published by Elsevier Ltd on behalf of ISA.

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

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C. Zhu, Z. Su, Y. Xia et al. / ISA Transactions xxx (xxxx) xxx

are obtained in [16], in which noises are coupled with each other. In addition, the state estimation dealing with correlated noises of the nonlinear systems is another important research field. In [17], it is showed that the two different estimator frameworks in [18] and [19] are equivalent for the nonlinear systems. For the similar noise-correlation assumption, Huang et al. [20] have presented a novel Gaussian approximate filter framework. In the wireless network systems, some common issues such as fadings, data transmission delays and packet losses, are inevitable due to the intrinsic properties of the network. These networkinduced problems severely limit the development of the state estimation and gain increasing research interest [21–23]. An optimal state estimator has been proposed in [24], and it is proved that the estimator is stable if the critical value of the arrival rate is within a certain interval. Further work has been studied in [25], where the weak convergence conditions are given for the system with intermittent observations. A distributed Kalman-consensus filter is designed for the systems suffering from unreliable communication links in [26], and the sufficient stochastic/uniform boundedness conditions of the system are given. In [27–29], the stability of the systems with different Markovian packet loss patterns (binary or bounded) has been considered. A related work has been found in [30], where the convergence conditions for the Kalman filter-based estimation are analyzed for the spatially distributed cyber–physical systems with a lossy network. For the multi-sensor systems, Sui et al. [31] have presented the stochastic stability conditions in the sense of least mean square. From the above statement, we know that the noise correlation and packet losses are widespread in the networked systems with limited bandwidth and constrained energy. However, as far as we know, there have been few discussions about state estimation considering event-triggered measurement transmission, noise correlation and packet losses simultaneously. In view of this, an event-triggered state estimator dealing with noise correlation and packet losses is designed in this paper. A communication mechanism that decides whether to send the measurement depending on a specific rule is presented to lessen the extra transmission. The correlation of the noises is decoupled by rewriting the state equation. Moreover, the packet loss process is modeled by a random variable. Then an event-triggered state estimator related to the communication rate, correlation coefficient and packet loss rate is presented. The major contributions are listed below: (i) A communication scheme which determines the measurements to be sent or not depending on a specific eventtriggered rule is presented to reduce the additional measurement transmissions, thus an expected trade-off between the rate of transmission and the estimator performance can be obtained through adjusting the trigger threshold properly. (ii) Based on the above strategy, a recursive event-triggered estimator related to the trigger threshold, the correlation coefficient and the packet loss rate is proposed for the networked systems suffering from correlated noises and unreliable communication links simultaneously, and the influence of the noise correlation and packet losses is weakened effectively. (iii) The performance of the state estimator is evaluated and the sufficient boundedness conditions of the proposed state estimator are given. The remainder of this paper is divided into five sections: Section 2 formulates the estimation problem suffering from noise correlation and packet losses. In Section 3, the event-triggered estimator related to the communication rate, correlation coefficient

and packet loss rate is derived. Section 4 analyzes the performance of the proposed estimator. Section 5 gives a simulation example to support the relevant results. Finally, we provide the conclusions and the recommendations of the future directions. Notation. Rn and Rm×n stand for the n-dimensional Euclidean space and set of all m × n matrices, respectively. For a symmetric matrix A, A ≥ 0 (A > 0) means that A is positive semi-definite (positive-definite). The relation A ≥ B (A > B) indicates that A − B ≥ 0 (A − B > 0). E {·} denotes the mathematical expectation. N (µ, σ 2 ) represents Gaussian distribution with mean µ and variance σ 2 . diag {a1 , a2 , ..., an } stands for a diagonal matrix with {a1 , a2 , ..., an } on the diagonal. tr(·) stands for the trace of a matrix, and ∥ · ∥ denotes the Euclidian norm of corresponding vectors or the spectral norm of corresponding matrices. I and 0 represent the identity matrix and the zero matrix with appropriate dimensions, respectively. The superscript T and −1 denote the transpose and inverse, respectively. 2. Problem formulation Consider the following linear discrete-time system: xk+1 = Axk + wk yk = Cxk + vk

(1) (2)

where xk ∈ Rn and yk ∈ Rm are the state and measurement vectors, respectively. A and C are system matrices, and (A, C ) is observable. The process noise wk ∈ Rn and the measurement noise vk ∈ Rm both satisfy zero-mean Gaussian distribution, and E {wk wlT } = Qk δkl , E {vk vlT } = Rk δkl , E {wk vlT } = Sk δkl where Qk ≥ 0, Rk > 0, Sk is the correlation coefficient, and

δkl =

{

1 0

if k = l , if k ̸ = l.

From the formula above, it is seen that wk and vk are crosscorrelated with E {wk vkT } = Sk . Moreover, the initial state x0 is Gaussian with mean xˆ 0|0 and covariance P0|0 , and is independent of wk and vk . Here a scenario of remote estimation is considered, where a communication strategy is presented to decide the measurement yk to be transferred or not depending on a specific event-triggered condition at each time instant k. Moreover, we assume that the communication link will suffer stochastic failures due to the unreliability of networks, which means that the measurements may be lost during the transmission. 2.1. Event-triggered mechanism Due to the constraint of the energy and the bandwidth in the wireless sensor networks, a communication strategy triggered by event is adopted to lessen the communication rate. A random variable γk is applied to model this process:

γk =

{

1 0

if (yk − y¯ k−1 )T (yk − y¯ k−1 ) > ρ, other w ise.

(3)

where y¯ k−1 represents the last measurement sent by the sensor before time instant k, and ρ is a given positive constant. From (3), it is seen that, the measurement yk will be sent to the remote estimator when γk = 1; Otherwise will not. Then the measurement used to update the state at time instant k is written as: y¯ k = γk yk + (1 − γk )y¯ k−1

(4)

It means that if γk = 1, then yk will be used to update the state. If γk = 0, only the measurement sent previously will be used.

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

C. Zhu, Z. Su, Y. Xia et al. / ISA Transactions xxx (xxxx) xxx

Remark 1. The event-triggered communication mechanism in (3) is essentially similar as that in [3–6]. Compared with the previous works [7–10,26], there is no information transmitted back to the sensor from the remote estimator, which can save the communication cost efficiently.

3

Theorem 1. Consider the linear discrete-time system described in (1)–(2) with the event-triggered communication strategy (3) and packet losses, the state estimation can be recursively executed as following: Time Update: xˆ k|k−1 = Axˆ k−1|k−1 + γk−1 λk−1 Jk−1 (yk−1 − C xˆ k−1|k−1 )

2.2. Packet loss

Pk|k−1

In this work, we suppose that the communication link is unreliable, and it means that the measurements may be lost during the transmission. To model this process, a random variable λk is applied referring to [24]: p(vk |λk ) =

{

N (0, Rk ) N (0, σ 2 I)

λk = 1 λk = 0

(5)

where σ → ∞. Note that λk = 1 means that the observation yk arrives normally, while yk is lost if λk = 0. The state estimation problem and the error covariance are defined as

(16)

= AP¯ k−1|k−1 AT + Qk−1 + γk−1 λk−1 × (Jk−1 C P¯ k−1|k−1 C T JkT−1 − Jk−1 Rk−1 JkT−1 − Jk−1 C P¯ k−1|k−1 AT − AP¯ k−1|k−1 C T JkT−1 )

(17)

Measurement Update: xˆ k|k = xˆ k|k−1 + γk Kk (yk − yˆ k|k−1 ) + (1 − γk )Lk (y¯ k − yˆ k|k−1 ) Kk = Pk|k−1 C (CPk|k−1 C + λk Rk + (1 − λk )σ I) T

T

2

−1

(18) (19)

Lk = (1 + α )Pk|k−1 C T [(1 + α )CPk|k−1 C T P¯ k|k

+ (1 + β )Rk + (1 + α −1 + β −1 )ρ I ]−1 = Pk|k−1 − γk Kk CPk|k−1 + (1 − γk ) × [(1 + α )(I − Lk C )Pk|k−1 (I − Lk C )T

(20)

xˆ k|k−1 ≜ E {xk |Yk−1 }

(6)

+ (1 + β )Lk Rk LTk + (1 + α −1 + β −1 )ρ Lk LTk − Pk|k−1 ]

xˆ k|k ≜ E {xk |Yk }

(7)

where yˆ k|k−1 = C xˆ k|k−1 , y¯ k = γk yk + (1 − γk )y¯ k−1 , Jk = Sk Rk−1 , σ → ∞ when λk = 0. The matrix P¯ k|k represents the upper bound of Pk|k defined in (9), i.e.,

T

Pk|k−1 ≜ E {(xk − xˆ k|k−1 )(xk − xˆ k|k−1 ) } Pk|k ≜ E {(xk − xˆ k|k )(xk − xˆ k|k )T }

(8) (9)

where Yk = (γ0 , ..., γk , λ0 , ..., λk , γ0 λ0 y0 , ..., γk λk yk ). The main task is to design a linear event-triggered estimator for the systems suffering from correlated noises and packet losses. 3. Event-triggered state estimation with correlated noises and packet losses In this part, a recursive event-triggered state estimator is derived to deal with the correlated noises and packet losses simultaneously. First, we will decouple the correlation between wk and vk . 1 Let xk+1 = Axk + wk + Jk (yk − Cxk − vk ), and Jk = Sk R− k , then xk+1 can be rewritten as xk+1 = A∗k xk + Jk yk + wk∗

(10)

where A∗k = A − Jk C

(11)

wk∗ = wk − Jk vk

(12)

Pk|k ≤ P¯ k|k

(21)

(22)

Proof. Considering the reliable and unreliable networks successively, we divide the proof into two steps. Step 1: Consider the event-triggered state estimation with reliable networks, i.e., the measurements are transmitted without packet loss. Let Yˇk = (γ0 , ..., γk , γ0 y0 , ..., γk yk ) and define xˇ k|k ≜ E {xk |Yˇk }, xˇ k|k−1 ≜ E {xk |Yˇk−1 },

(23)

Pˇ k|k ≜ E {(xk − xˇ k|k )(xk − xˇ k|k ) }, Pˇ k|k−1 ≜ E {(xk − xˇ k|k−1 )(xk − xˇ k|k−1 )T }

(24)

T

First, the time update is considered. If γk−1 = 1, from (10) we have xˇ k|k−1 = Axˆ k−1|k−1 + Jk−1 (yk−1 − C xˆ k−1|k−1 )

(25)

Pˇ k|k−1 = E {(A∗k−1 x¯ k−1|k−1 + wk−1 − Jk−1 vk−1 )

× (A∗k−1 x¯ k−1|k−1 + wk−1 − Jk−1 vk−1 )T } = A∗k−1 Pˇ k−1|k−1 (A∗k−1 )T + Qk−1 − Jk−1 SkT−1

(26)

where x¯ k−1|k−1 = xk − xˇ k−1|k−1 . If γk−1 = 0, then

From (12), we have

xˇ k|k−1 = Axˆ k−1|k−1

(27)

E {wk (wl ) } = Qk δkl

(13)

Pˇ k|k−1 = APˇ k−1|k−1 AT + Qk−1

(28)

E {wk vl } = 0

(14)

∗

∗ T

∗

∗ T

1 where Qk∗ = Qk − Sk R− k Sk . Hence, the correlation between the noises is decoupled in the equivalent system (10) and (2). Next, we will propose a recursive event-triggered estimation algorithm to deal with noise correlation and packet losses. First, a lemma is introduced below:

Lemma 1 ([32]). For any two vectors a, b ∈ Rn , the following inequality holds: ab + ba ≤ ε aa + ε T

T

T

where ε is a scalar.

−1

T

bb

(15)

Combining (25)–(28), we have xˇ k|k−1 = Axˆ k−1|k−1 + γk−1 Jk−1 (yk−1 − C xˆ k−1|k−1 ) Pˇ k|k−1 = APˇ k−1|k−1 A + Qk−1 + γ T

−

Jk−1 Rk−1 JkT−1

(29)

T T k−1 (Jk−1 C Pk−1|k−1 C Jk−1

ˇ

− Jk−1 C Pˇ k−1|k−1 AT − APˇ k−1|k−1 C T JkT−1 )(30)

Now the measurement update is considered. If γk = 1, the measurement update degenerates to standard Kalman filter, and the following equations can be obtained intuitively: xˇ k|k = xˇ k|k−1 + Kˇ k (yk − yˇ k|k−1 )

(31)

yˇ k|k−1 = C xˇ k|k−1

(32) T

T

Pˇ k|k = Pˇ k|k−1 − Pˇ k|k−1 C (C Pˇ k|k−1 C + Rk )

−1

C Pˇ k|k−1

(33)

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

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Kˇ k = Pˇ k|k−1 C T (C Pˇ k|k−1 C T + Rk )−1

(34)

When γk = 0, let xˇ k|k = xˇ k|k−1 + Lˇ k (y¯ k − yˇ k|k−1 )

(35)

Then we can rewrite the above equation below: xˇ k|k = xˇ k|k−1 + Lˇ k (yk − yˇ k|k−1 ) + Lˇ k (y¯ k−1 − yk )

(36)

Subtracting (36) from (1), the estimation error is obtained: x¯ k|k = (I − Lˇ k C )x¯ k|k−1 − Lˇ k vk − Lˇ k (y¯ k−1 − yk )

(37)

Following the above discussion, we can yield the eventtriggered state estimator with reliable networks: xˇ k|k−1 = Axˆ k−1|k−1 + γk−1 Jk−1 (yk−1 − C xˆ k−1|k−1 )

¯ k−1|k−1 AT + Qk−1 + γk−1 (Jk−1 C P¯ k−1|k−1 C T JkT−1 Pˇ k|k−1 = AP − Jk−1 Rk−1 JkT−1 − Jk−1 C P¯ k−1|k−1 AT − AP¯ k−1|k−1 C T JkT−1 ) xˇ k|k = xˇ k|k−1 + γk Kˇ k (yk − yˇ k|k−1 ) + (1 − γk )Lˇ k (y¯ k − yˇ k|k−1 ) Kˇ k = Pˇ k|k−1 C T (C Pˇ k|k−1 C T + Rk )−1 Lˇ k = (1 + α )Pˇ k|k−1 C T [(1 + α )C Pˇ k|k−1 C T

+ (1 + β )Rk + (1 + α −1 + β −1 )ρ I]−1

Then we have

P¯ k|k = Pˇ k|k−1 − γk Kˇ k C Pˇ k|k−1

Pˇ k|k = E {¯xk|k x¯ Tk|k }

+ (1 − γk )[(1 + α )(I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T + (1 + β )Lˇ k Rˇ k Lˇ Tk + (1 + α −1 + β −1 )ρ Lˇ k Lˇ Tk − Pˇ k|k−1 ]

= (I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T + Lˇ k Rk Lˇ Tk + Lˇ k E {(y¯ k−1 − yk )(y¯ k−1 − yk )T }Lˇ Tk − (I − Lˇ k C )E {¯xk|k−1 (y¯ k−1 − yk )T }Lˇ Tk − Lˇ k E {(y¯ k−1 − yk )x¯ Tk|k−1 }(I − Lˇ k C )T + Lˇ k E {vk (y¯ k−1 − yk )T }Lˇ Tk + Lˇ k E {(y¯ k−1 − yk )vkT }Lˇ Tk

(38)

According to the event-triggered communication mechanism (3) and Lemma 1, we can get the following inequalities:

xˆ k|k−1 = Axˆ k−1|k−1 + γk−1 λk−1 Jk−1 (yk−1 − C xˆ k−1|k−1 ) Pk|k−1 = AP¯ k−1|k−1 AT + Qk−1 + γk−1 λk−1

−(I − Lˇ k C )E {¯xk|k−1 (y¯ k−1 − yk )T }Lˇ Tk −Lˇ k E {(y¯ k−1 − yk )x¯ Tk|k−1 }(I − Lˇ k C )T ≤ α (I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T + α −1 Lˇ k E {(y¯ k−1 − yk )(y¯ k−1 − yk )T }Lˇ Tk ≤ α (I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T + α −1 ρ Lˇ k Lˇ Tk (39) and

× (Jk−1 C P¯ k−1|k−1 C T JkT−1 − Jk−1 Rk−1 JkT−1 − Jk−1 C P¯ k−1|k−1 AT − AP¯ k−1|k−1 C T JkT−1 ) xˆ k|k = xˆ k|k−1 + γk Kk (yk − yˆ k|k−1 ) + (1 − γk )Lk (y¯ k − yˆ k|k−1 ) Kk = Pk|k−1 C T (CPk|k−1 C T + λk Rk + (1 − λk )σ 2 I)−1 Lk = (1 + α )Pk|k−1 C T [(1 + α )CPk|k−1 C T

Lˇ k E {vk (y¯ k−1 − yk )T }Lˇ Tk + Lˇ k E {(y¯ k−1 − yk )vkT }Lˇ Tk ≤ β Lˇ k Rk Lˇ Tk + β −1 Lˇ k E {(y¯ k−1 − yk )(y¯ k−1 − yk )T }Lˇ Tk ≤ β Lˇ k Rk Lˇ Tk + β −1 ρ Lˇ k Lˇ Tk

P¯ k|k (40)

where α and β are positive scalars. An upper bound of Pˇ k|k is obtained by substituting (39) and (40) into (38):

P¯ k|k = (1 + α )(I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T

+ (1 + β )Lˇ k Rk Lˇ Tk + (1 + α −1 + β −1 )ρ Lˇ k Lˇ Tk

(41)

The estimator gain Lˇ k can be designed by solving the partial derivative equation:

∂ tr(P¯ k|k ) =0 ∂ Lˇ k Then Lˇ k = (1 + α )Pk|k−1 C T [(1 + α )CPk|k−1 C T

+ (1 + β )Rk + (1 + α −1 + β −1 )ρ I]−1

(42)

Combining (31), (33)–(35), (41) and (42), the measurement update with reliable networks is presented as follows: xˇ k|k = xˇ k|k−1 + γk Kˇ k (yk − yˇ k|k−1 ) + (1 − γk )Lˇ k (y¯ k − yˇ k|k−1 ) Kˇ k = Pˇ k|k−1 C T (C Pˇ k|k−1 C T + Rk )−1 Lˇ k = (1 + α )Pˇ k|k−1 C T [(1 + α )C Pˇ k|k−1 C T

P¯ k|k

Step 2: Now consider the event-triggered state estimation with unreliable networks. Note that packet loss happens only during the measurement transmission (γk = 1), then motivated by [24], an event-triggered estimator dealing with the correlated noises and packet losses is given intuitively:

+ (1 + β )Rk + (1 + α −1 + β −1 )ρ I]−1 = Pˇ k|k−1 − γk Kˇ k C Pˇ k|k−1 + (1 − γk ) × [(1 + α )(I − Lˇ k C )Pˇ k|k−1 (I − Lˇ k C )T + (1 + β )Lˇ k Rˇ k Lˇ Tk + (1 + α −1 + β −1 )ρ Lˇ k Lˇ Tk − Pˇ k|k−1 ]

+ (1 + β )Rk + (1 + α −1 + β −1 )ρ I]−1 = Pk|k−1 − γk Kk CPk|k−1 + (1 − γk ) × [(1 + α )(I − Lk C )Pk|k−1 (I − Lk C )T + (1 + β )Lk Rk LTk + (1 + α −1 + β −1 )ρ Lk LTk − Pk|k−1 ]

1 where yˆ k|k−1 = C xˆ k|k−1 , y¯ k = γk yk + (1 − γk )y¯ k−1 , Jk = Sk R− k , σ → ∞ when λk = 0. □

Remark 2. Since the event-triggered system with packet losses is not necessarily Gaussian, it is hard to obtain the optimal state estimation. To reduce the effect of the transmission scheme introduced previously and the packet losses, a suboptimal state estimator with a lower bandwidth requirement and energy consumption is proposed in Theorem 1. Remark 3. ρ is the trigger threshold of the data transmission mechanism, and it can be set according to the actual physical system and the accuracy requirement, etc. In addition, the parameters α and β are given constants that can be selected ¯ k|k . appropriately to adjust the upper bound P Remark 4. Compared with [3,9–11,26], this work investigates the event-triggered estimation problem with noise correlation and packet losses simultaneously. Meanwhile, the upper bound P¯ k|k varies with the threshold ρ , which means that a tradeoff between the rate of transmission and the estimator performance can be obtained through co-design of the data transmission mechanism and the estimator. 4. Performance analysis The estimator performance will be evaluated and certain boundedness conditions for the covariance expectation will be given in this part.

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

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5

To accomplish the subsequent developments, a useful lemma is given before presenting the main results.

First, consider the case when k = 2. From (50), we have

Lemma 2 ([21]). If A, B ∈ Rn×n satisfy A > 0 and B > 0, then

+ {q + (1 − γ )a2 c −2 [(1 + β )r + (1 + α −1 + β −1 )ρ] ¯ + γ λa2 c 2 r − 2γ λacs}I ≤ γ (1 − λ)a2 pI + pI ∑k−2 Next, assume E {Pk−1|k−2 } ≤ p i=0 {γ (1 − λ)a2 }i I, then

(A + B)−1 > A−1 − A−1 BA−1

(43)

Theorem 2. Consider the linear discrete-time system described in (1)–(2) with the event-triggered communication strategy (3) and packet losses, if there exist real constants a, c , s, q, r and ρ¯ > 0 such that

∥ A ∥≤ a, c ≤∥ C ∥, ∥ Sk ∥≤ s, Qk ≤ qI , Rk ≤ rI , ρ ≤ ρ¯ and if λ > 1 −

1

γ a2

E {P2|1 } ≤ γ (1 − λ)AE {P1|0 }AT

E {Pk|k−1 } ≤ γ (1 − λ)AE {Pk−1|k−2 }AT

+ {q + (1 − γ )a2 c −2 [(1 + β )r + (1 + α −1 + β −1 )ρ] ¯ + γ λa2 c 2 r − 2γ λacs}I k−2 ∑ ≤ γ (1 − λ)a2 p {γ (1 − λ)a2 }i I + pI

(44) (45)

and C is nonsingular, then

i=0 k−1

E {Pk|k−1 } ≤ p¯ I

(46)

If λ > 1 −

P¯ k|k = Pk|k−1 − γk λk Pk|k−1 C T (CPk|k−1 C T + Rk )−1 CPk|k−1

(47)

Substituting (47) into (17), the following derivation is derived: Pk|k−1 = (1 − γk−1 λk−1 )AP¯ k−1|k−1 AT + Qk−1 1 T × (A − Jk−1 C )T − Sk−1 R− k−1 Sk−1 ]

1 T + Qk−1 − γk−1 λk−1 Sk−1 R− k−1 Sk−1

+ (1 − γk−1 )AC −1 [(1 + β )Rk−1 + (1 + α −1 + β −1 )ρ I ]C −T AT 1 −1 + γk−1 λk−1 (A − Sk−1 R− Rk−1 C −T k−1 C )C

[

+ (1 − γ )AC −1 [(1 + β )Rk−1 + (1 + α −1 + β −1 )ρ I ]C −T AT 1 −1 + γ λ(A − Sk−1 R− Rk−1 C −T k−1 C )C (49)

Combining (44) and (45), it is obtained that E {Pk|k−1 } ≤ γ (1 − λ)AE {Pk−1|k−2 }AT

+ {q + (1 − γ )a2 c −2 [(1 + β )r + (1 + α −1 + β −1 )ρ] ¯ 2 2 + γ λa c r − 2γ λacs}I (50) Let p = max{∥ E {P1|0 } ∥, q + (1 − γ )a2 c −2 [(1 + β )r + (1 + α −1 + β −1 )ρ] ¯ + γ λa2 c 2 r − 2γ λacs}, then it is recursively proved k−1 ∑ {γ (1 − λ)a2 }i I i=0

0.9 0

□

(51)

0.1 0.9

]

x(k − 1) + w (k − 1)

0 ]x(k) + v (k)

y(k) = [ 1 (48)

1 T E {Pk|k−1 } ≤ γ (1 − λ)AE {Pk−1|k−2 }AT + Qk−1 − γ λSk−1 R− k−1 Sk−1

E {Pk|k−1 } ≤ p

, the sum will converge and (46) is satisfied.

Remark 5. From Theorem 2, it is seen that the boundedness of the proposed estimator can be guaranteed through adjusting the communication rate γ which is related to the threshold ρ . However, higher transmission rate means more energy consumption and bandwidth requirement, thus an expected compromise can be achieved between the resource consumption and the estimator performance by adjusting ρ . Moreover, a critical value 1 − γ 1a2 is obtained to guarantee the estimator boundedness, and it implies that the packet loss rate γ should be limited with an upper bound.

x(k) =

Define γ = E {γk } as the average sensor communication rate and λ = E {λk } as the arrival probability of the transmitted measurement, we have

1 T × (A − Sk−1 R− k−1 C )

1

γ a2

In this part, we will introduce an example to support the previously relevant results. The following target tracking system with same parameters as in [9] is considered:

T

1 T × (A − Sk−1 R− k−1 C )

(52)

5. Simulation

+ γk−1 λk−1 [(A − Jk−1 C )P¯ k−1|k−1 ≤ γk−1 (1 − λk−1 )APk−1|k−2 A

∑ {γ (1 − λ)a2 }i I i=0

Proof. First, we substitute (19) and (20) into (21):

+ (1 − γk ){α Pk|k−1 − (1 + α )2 × Pk|k−1 C T [(1 + α )CPk|k−1 C T + (1 + β )Rk + (1 + α −1 + β −1 )ρ I]−1 CPk|k−1 } ≤ Pk|k−1 − γk λk (Pk|k−1 − C −1 Rk−1 C −T ) + (1 − γk ){C −1 [(1 + β )Rk + (1 + α −1 + β −1 )ρ I]C −T − Pk|k−1 }

=p

where v (k) = [1 1]w (k) + ϑ (k), w (k) and ϑ (k) are independent Gaussian noises with expectation E {w (k)} = 0, E {ϑ (k)} = 0, covariance Qk = [1 0; 0 1] and Θk = 0.01, respectively. Then the covariance of v (k) is Rk = 2.01. Meanwhile, w (k) is coupled with v (k) and the correlation coefficient is Sk = [1 1]T . The initial conditions are xˆ 0|0 = [1 1]T and P0|0 = [1 0; 0 1]. α and β are set ∑M as 0.2. Let MSEi (i = 1, 2) = (1/M) k=1 (xi,k − xˆ i,k|k )2 represent the mean-square error (MSE) of the ith component of x, where M is the sample size. The transmission rate γ is calculated as [7]:

γ =

N 1 ∑

N

{γk }

(53)

k=1

Without considering noise correlation and data dropouts, the MSEs of the estimator in this paper and [1] are shown in Figs. 1– 2. It is seen that the estimator of this paper performs better due to the co-design of the estimator and the communication mechanism. In Figs. 3–4, the comparison results of the MSEs are given under the worse system environment (i.e., the noise correlation and packet losses are considered). It is showed that, compared with the significant deterioration of the estimation performance in [1], the proposed estimator can weaken the influence of correlation and packet losses obviously.

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

6

C. Zhu, Z. Su, Y. Xia et al. / ISA Transactions xxx (xxxx) xxx

Fig. 1. MSE1 of the state estimation without considering noise correlation and packet losses.

Fig. 4. MSE2 of the state estimation considering noise correlation and packet losses.

Fig. 2. MSE2 of the state estimation without considering noise correlation and packet losses.

Fig. 5. The values of γk .

slightly. However, the trace increases rapidly when ρ = 10. It means that the performance will deteriorate severely if the trigger threshold is too large. Fig. 7 reveals the relationship of ρ and γ , where γ decreases with the increasing of ρ . It is indicated that an expected trade-off between the rate of transmission and the estimator performance can be obtained through tuning the threshold ρ properly. The traces of the covariance with different correlation coefficient S and packet loss rate λ are presented in Figs. 8–9, from which we can see that the performance will deteriorate while increasing the correlation coefficient or the packet loss rate. Meanwhile, Figs. 6 and 9 show that the trace of the covariance is bounded when the trigger threshold and the packet loss rate are small enough. 6. Conclusions Fig. 3. MSE1 of the state estimation considering noise correlation and packet losses.

From Figs. 5–6, it is seen that, when the trigger threshold

ρ = 0.2, the communication rate γ decreases by 15% compared with ρ = 0 (γ = 1), while the trace of the covariance just changes

In this work, an event-triggered estimation algorithm dealing with the correlated noises and packet losses simultaneously has been proposed. A communication mechanism determines whether to send the measurement yk depending on a specific event-triggered condition is introduced. Based on that, a recursive event-triggered state estimator has been presented. It is beneficial for the networks with constrained energy and bandwidth

Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.

C. Zhu, Z. Su, Y. Xia et al. / ISA Transactions xxx (xxxx) xxx

Fig. 6. The traces of the covariance of the estimator with different ρ (S = [1 1]T ,

λ = 0.1).

Fig. 7. The communication rate γ versus ρ .

and the influence of the noise correlation and packet losses is weakened. Moreover, the sufficient boundedness conditions for the estimator have been obtained. At last, a numerical example is illustrated to support the relevant algorithms. An interesting future direction is to extend the previous method to the multi-channel systems.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments The authors thank the reviewers for their valuable and helpful comments which have improved the presentation. The work was supported by the National Natural Science Foundation of China (61603047, 61773334, 61703040), the Scientific Research Project of Beijing Municipal Educational Commission (KM201911232014).

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Fig. 8. The traces of the covariance of the estimator with different S (ρ = 0.2, λ = 0.1).

Fig. 9. The traces of the covariance of the estimator with different λ (ρ = 0.2, S = [1 1]T ).

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Please cite this article as: C. Zhu, Z. Su, Y. Xia et al., Event-triggered state estimation for networked systems with correlated noises and packet losses. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.038.