Event-based finite horizon state estimation for stochastic systems with network-induced phenomena

Abstract

This paper investigates a finite horizon state estimation problem for a class of discrete-time stochastic systems with random transmission delays and out-of-order packets of data. Employing an event-driven signal-choosing scheme of logic zero-order-holder (LZOH), a system model is established synthetically in a unified form considering the network-induced phenomena, to drop out-of-order packets and improve system performance. By virtue of the established system model, a novel minimum error covariance matrix for the augmented state-space is obtained from the estimated variance constraint. With the aid of a finite horizon, the upper boundary of estimation error covariance is introduced during the information transmission from sensor to estimator, and the appropriate filter parameters are probed. To improve the estimation performance and alleviate the computation burden, an estimation-based compensation approach for random transmission delays is proposed using the received valid signals. Finally, the effectiveness and applicability of the proposed state estimation method are illustrated by a numerical simulation.

Keywords

finite horizon state estimation out-of-order packets stochastic systems transmission delays

Introduction

With the rapid development of complex networked systems, the technology of information awareness and computation in the context of complex systems has become a focus of research due to its successful application in many areas, such as cyber-physical systems (CPSs) (Reppa et al. 2015; Shi et al. 2016), smart grids (Yu et al. 2015), communication networks (Ge and Han 2015; Liu et al. 2015) and so on. Due to the limited bandwidth and complex communication resources, in order to suit the growing use of information computation, it is essential to design a communication scheme and lower installation and maintenance costs (Han et al. 2017; Hu et al. 2016; Peng et al. 2013; Reppa et al. 2015). However, the limited capacity of the communication bandwidth leads to various network-induced phenomena, such as transmission delays, stochastic parameters, missing measurements, communication link failures and so on, which usually negatively impact the system performance (Sun et al. 2014). Therefore, it is significant to study the state estimation problem with varied network-induced phenomena.

Some interesting results have been found for stochastic systems with various disturbances for noise. Multiplicative noise (Feng et al. 2013; Liu et al. 2014; Tian et al. 2016; Wang et al. 2016) is used to describe the parameter of stochastic uncertainties. You et al. (2017) improve set-membership guaranteed state estimation for uncertain non-linear systems with an unknown but bounded description of disturbance and noise. In engineering applications, noise disturbances are generated by communication facilities and channels. Hence, a correlation between the process noise and measurement noise is involved. Based on the correlated noise, in fractional-order systems, the state estimation with coloured measurement noise is presented, which converts the coloured measurement noise to white measurement noise (Safarinejadian et al. 2017). Taking into account the correlation, many estimation methods are presented for networked systems, such as the cross-correlation between the measurement noise and process noise as discussed by Feng et al. (2013) and Yan et al. (2013). Liu et al. (2017) present a weighted reorganized approach for innovation and error cross-covariance matrices, and investigate a distributed robust Kalman filtering issue with data transmission time-delay and cross-correlated noises. For noise sequences with uncertain variances, the actual filtering error variances (Qi et al. 2014; Zhang et al. 2015) are obtained with a minimal upper boundary for all admissible uncertainties. Note that noise processing for stochastic systems increases the system complexity and deteriorates the system performance.

To reduce energy consumption and extend the lifetimes of communications, an event-driven strategy is presented. Using a minimum mean-square error estimator, Shi et al. (2016) investigate an event-based state estimation problem for linear-time-varying systems with unknown inputs. Liu et al. (2017) design the finite horizon quantized filter for a class of time-varying systems with quantization effects, and a componentwise event-triggered transmission strategy is proposed. Considering the measurement losses, the event-based distributed recursive filtering problem is investigated for discrete-time state-saturated systems (Wen et al. 2018). For distributed state estimation problems, Chen et al. (2017) discuss a time-variant model with varying delays, random non-linearity and external disturbances; moreover, a novel event-triggered robust state estimation is proposed. In Kalman filter applications, a finite horizon Gaussianity-preserving event-based sensor scheduling (Wu et al. 2016) is designed. However, the event handling increases energy consumption and computational complexity.

Taking into account that random transmission delays inevitably generate packet disorders, the typical signal-choosing scheme of logic zero-order-holder (LZOH) is used for dropping the network-induced out-of-order packets phenomenon. The LZOH scheme employs an event-driven mechanism, and develops a function to judge the ordered or out-of-order data packets before being transmitted. Then, the LZOH only receives the latest timestamped data packet (Ge and Han 2015; Peng et al. 2013, 2014; Wang et al. 2015; Zhang and Han 2013). Therefore, the LZOH has the capability of choosing the most recently arrived data packet and dropping other data packets. Meanwhile, the packet disorders are actively discarded. The LZOH scheme is widely applied in networked control systems for the event-trigger mechanism. However, analysing and designing a finite horizon state estimation based on the signal-choosing scheme is more complex.

For measurements of transmission via unreliable communication channels, the issue of state estimation or filtering has attracted great interest. Yang et al. (2016) investigate a robust adaptive state estimation for uncertain non-linear switched systems with unknown inputs. The augmented state approach (Chen et al. 2014a; Wang et al. 2015) applies the compensation scheme by one-step prediction to describe the random delays. To transform the random-delayed system into a delay-free one, the measurement reorganization approach (García et al. 2015; Rezaei et al. 2015) is an effective strategy. In the work of Sun et al. (2014), Wang et al. (2016),Wang et al. (2015) and Sun and Xiao (2013), the authors employ the state augmentation strategy to describe packet drop-outs, and the system is transferred into random variables of Bernoulli distribution. For finite horizon filtering (Chen et al. 2014b; Rezaei et al. 2015), because the actual error covariance of the estimated state is less than the upper boundary, the finite horizon filter has better transient performance for filtering processes in networked systems. Han et al. (2017) propose the distributed finite horizon $H \infty$ -consensus control for time-varying multi-agent systems with random parameters. To deal with high-rate communication networks and stochastic protocols, $H \infty$ finite horizon filtering is proposed for developing mapping technologies (Zou et al. 2017). Chen et al. (2017) propose finite horizon $H \infty$ filtering for switched time-varying stochastic systems with random sensor non-linearities and packet drop-outs. From the existing literature, it is difficult to design for state estimation with random transmission delays and out-of-order packets; the difficulties include dealing with the disordered and missing data packets, as well as probing the appropriate filter parameters.

Motivated by the above discussion, this paper focuses on designing an event-based finite horizon state estimation for stochastic systems via network-induced random transmission delays and out-of-order packets. The main contributions of this paper are summarized as follows:

The system model employing the LZOH signal-choosing scheme is established in a unified form. The LZOH scheme is widely applied in networked control systems (Ge and Han 2015; Peng et al. 2013, 2014; Wang et al. 2015; Zhang and Han 2013). The proposed approach designs a finite horizon state estimation based on the event-driven mechanism, which is used for dropping out-of-order packets from sensor to processor, and improving system performance.

Employing the augmented state-space with reorganized data packets, the upper boundary of estimation error covariance is obtained from the estimated variance constraint. It is different from employing the state augmentation strategy of Bernoulli distribution (Sun and Xiao 2013; Sun et al. 2014; Wang et al. 2015, 2016), and the augmented state-space with reorganized measurement is an effective strategy to probe the appropriate filter parameters.

An estimation-based compensation approach for random transmission delays is proposed using the received valid signals with time-stamp. To describe the random delays, different from the augmented state approach (Chen et al. 2014a; Wang et al. 2015) of applying the compensation scheme by one-step prediction, the estimation-based compensation improves the estimation performance and alleviates the computational burden.

The remainder of this paper is organized as follows. The stochastic system modelling and addressed problems are described for random transmission delays and out-of-order packets in the next section. Then, we design the event-based finite horizon state estimation using the LZOH scheme. We verify the effectiveness and the applicability of the proposed approach in the following section, and provide concluding remarks at the end of the paper.

Problem formulation

System description

Stochastic systems are widely applied in engineering (Feng et al. 2013; Rezaei et al. 2015; Tian et al. 2016; Wang et al. 2015), and the state and measurement equations of sensors are described by the following linear discrete-time system (Tian et al. 2016;Wang et al. 2016):

x_{k + 1} = (A_{k} + F_{k} F_{k} E_{k}) x_{k} + B_{k} w_{k}, k = 1, 2, \dots

(1)

z_{k} = (C_{k} + H_{k} F_{k} E_{k}) x_{k} + v_{k},

(2)

where $x_{k} \in R^{r}$ represents the system state to be estimated, $z_{k} \in R^{m}$ expresses the measurement output at the time instant of k. The parameter uncertainty of the stochastic system is denoted by the multiplicative noise $F_{k}$ . Note that $F_{k}$ satisfies $F_{k} F_{k}^{T} ⩽ I$ . Then, $w_{k} \in R$ and $v_{k} \in R^{m}$ are zero-mean white noises with covariances $Q_{k}$ and $R_{k}$ , respectively. $A_{k} \in R^{r \times r}$ , $B_{k} \in R^{r}$ , $C_{k} \in R^{m \times r}$ , $F_{k}$ , $H_{k}$ and $E_{k}$ are known time-varying matrices. The initial state $x_{0}$ with mean $μ_{0}$ and covariance $P_{0}$ , which is assumed to be uncorrelated with other noise signals.

Due to the influence of communication constraints, the correlated noise (Tian et al. 2016; Wang et al. 2015, 2016) for the dynamical systems is dealt with in this paper. Assuming that the process noise $w_{k}$ and measurement noise $v_{k}$ are correlated at the same time instant, the statistical properties satisfy as follows:

\begin{array}{l} E (w_{k}) = 0, E (v_{k}) = 0, \\ E ((\begin{array}{l} w_{k} \\ v_{k} \end{array}) (w_{l}^{T} {(v_{l})}^{T})) = (\begin{array}{l} Q_{k} δ_{k, l} S_{k} δ_{k, l} \\ {(S_{k})}^{T} δ_{k, l} R_{k} δ_{k, l} \end{array}), \end{array}

(3)

where $δ_{k, l}$ denotes the Kronecker function, and $Q_{k} = Q_{k}^{T}$ , $R_{k} = R_{k}^{T}$ , $S_{k} = S_{k}^{T}$ .

System modelling

Taking into account the network-induced random transmission delays and out-of-order packets, Figure 1 shows the analysis of the process of state estimation using the LZOH scheme. It is worth noticing that the measurement $z_{k}$ is timestamped before being transmitted, (i.e. $z_{1}, z_{2}, \dots$ are the transmitted data packets with orderly labelling). Since the LZOH scheme develops a function to judge the latest timestamped data packet. During the transmission of measurement $z_{k}$ , the event-driven LZOH scheme stores the latest data packets, which will be updated when the data packet receives a later signal.

A typical scenario is shown in Figure 2. We set the constant sampling period as T, and the sampling time instant is $t \in {kT, k \in N}$ . Suppose that the largest delay is no more than N. For simplicity, set the maximum of time delay be $N = 5$ , and $τ_{k}$ represents the transmission delays at the time instant of k. An example of the sampled packets is given in Figure 2, which includes three cases:

Case 1 When $t \in [2 T, 5 T)$ , the time-stamped signals $z_{2}$ and $z_{3}$ are transmitted, and the random time-delays are represented as $η (t_{2})$ and $η (t_{3})$ , respectively. As shown in Figure 2, $η (t_{2}) > η (t_{3})$ , so $z_{2}$ and $z_{3}$ generate the out-of-order packets during the transmission. With the aid of the judgement function for the LZOH scheme, $z_{3}$ is the latest signal, and signal $z_{2}$ is discarded.

Case 2 At the sampling time $9 T$ and $10 T$ , $z_{7}$ is dropped at the time instant of 10 using the LZOH scheme, at the same time, the latest time-stamped data packet $z_{8}$ is stored.

Case 3 When $t \in [12 T, 14 T)$ , the received valid data packet is $z_{11}$ , so that $z_{10}$ and $z_{9}$ are discarded from the judgement of the latest data packet.

Note that the latest data packet before being transmitted is close to the current actual signal to be estimated; the packet disorders are dropped actively via the LZOH scheme (Peng et al. 2013, 2014; Zhang and Han 2013).

Remark 1. It is assumed that the current time instant is k, and the transmission delay is $τ_{k}$ . The received valid data packet with a time-stamp is denoted by $k_{1}$ , adopting the LZOH scheme. The artificial transmission delay from sensor to processor is represented by $τ_{k_{1}}$ using an event-driven mechanism, and $τ_{k_{1}} \in N$ denotes the transmission delay at the sampling time instants. They have the equivalence relation:

k = τ_{k_{1}} + k_{1} .

(4)

When the LZOH receives the valid data packet $z_{k_{1}}$ before being transmitted, the stored signal $y_{k}$ from (4) is modelled as

y_{k} = z_{k_{1}} .

(5)

It is worth noticing that the received valid data packets are modelled synthetically with transmission delays and out-of-order packets by the signal-choosing scheme of LZOH. Meanwhile, the time-stamped data packet implies that the processor is able to obtain the knowledge of the data delays and dropout packets during the transmission (Rezaei et al. 2015).

For the received valid sequence ${y_{0}, \dots, y_{k - 1}, y_{k}}$ with random delays and out-of-order packets, the purpose of state estimation is translated into probing the minimum error covariance estimation ${\hat{x}}_{k | k}$ of $x_{k}$ for the LZOH scheme, which is compensated by the estimation ${\hat{x}}_{k_{1} | k_{1}}$ given in equations (4) and (5).

Event-based finite horizon state estimation for LZOH

A finite horizon state estimation approach based on the event-driven mechanism is presented in this section. First of all, the objective of the finite horizon state estimation is to obtain a guaranteed upper boundary using the minimum estimation error covariance. Then, the linear matrix inequalities of Lemmas 2 and 3 are introduced.

Lemma 2. Rezaei et al. (2015) assumed that matrices A, H, E and F have compatible dimensions, such that $F F^{T} \leq I$ . Let X be a symmetric positive definite matrix, and $α > 0$ be an arbitrary positive constant. The inequality $α^{- 1} I - EX E^{T} > 0$ is satisfied. Then, the following inequality holds:

(A + HFE) X {(A + HFE)}^{T} ⩽ A {(X^{- 1} - α E^{T} E)}^{- 1} A^{T} + α^{- 1} H H^{T},

(6)

in which ${(X^{- 1} - α E^{T} E)}^{- 1} = X + X E^{T} {(α^{- 1} I - EX E^{T})}^{- 1} EX$ is obtained based on the matrix inversion lemma.

Lemma 3. (Rezaei et al. 2015) The sampling time instants are denoted by t and k, which satisfy $0 \leq t \leq k$ . It is assumed that X and Y are symmetric positive definite matrices, and subject to $s_{t} (X) = s_{t}^{T} (X) \in R^{n \times n}$ and $h_{t} (X) = h_{t}^{T} (X) \in R^{n \times n}$ . If there exists $Y > X$ , such that $s_{t} (Y) \geq s_{t} (X)$ and $h_{t} (Y) \geq s_{t} (Y)$ , then the solutions $M_{t}$ and $N_{t}$ in terms of the following recursive equations:

M_{t + 1} = s_{t} (M_{t}), N_{t + 1} = h_{t} (N_{t}), M_{0} = N_{0} > 0

(7)

satisfy $M_{t} ⩽ N_{t}$ .

Suppose that the current sampling time instant is k, and the stored data packet is $y_{k}$ , that is, the received data packet is $z_{r}$ before being transmitted with artificial delay $τ'_{k}$ . Set $r = k - τ'_{k}$ , where $k_{1}$ and $τ_{k_{1}}$ in equation (4) are redefined as r and $τ'_{k}$ , respectively. The stored measurement is then reorganized from equations (2) and (5):

y_{k} = z_{r} = (C_{r} + H_{r} F_{r} E_{r}) x_{r} + v_{r} .

(8)

Note that the state estimator possesses enough processing capability to compute the optimal state estimate ${\hat{x}}_{r | r}$ depending on the stored data packets ${y_{0}, \dots, y_{k - 1}, y_{k}}$ . Therefore, the structure of the state estimation based on Kalman-type filtering is designed with time-stamped signals:

{\hat{x}}_{r | r} = {\hat{x}}_{r | r - 1} + K_{r} (z_{r} - {\hat{C}}_{r} {\hat{x}}_{r | r - 1}),

(9)

{\hat{x}}_{r + 1 | r} = {\hat{A}}_{r} {\hat{x}}_{r | r - 1} + L_{r} (z_{r} - {\hat{C}}_{r} {\hat{x}}_{r | r - 1}),

(10)

where ${\hat{C}}_{r}$ , $K_{r}$ , ${\hat{A}}_{r}$ and $L_{r}$ are filter parameters.

Upper boundary for estimation error covariance

To obtain the solutions of the upper boundaries from the filtering and prediction covariance matrices, the augmented state vectors from the reorganized system model in equations (1) and (8) are defined as follows:

{\tilde{Ψ}}_{r} = [\begin{matrix} {\tilde{e}}_{r} \\ {\hat{x}}_{r | r - 1} \end{matrix}], Ψ_{r} = [\begin{matrix} e_{r} \\ {\hat{x}}_{r | r} \end{matrix}] .

(11)

Note that ${\tilde{e}}_{r} \overset{Δ}{=} x_{r} - {\hat{x}}_{r | r - 1}$ expresses the prediction error, and $e_{r} \overset{Δ}{=} x_{r} - {\hat{x}}_{r | r}$ is the filtering error.

Then, denote ${\tilde{Ψ}}_{r + 1} = [\begin{matrix} {\tilde{e}}_{r + 1} \\ {\hat{x}}_{r + 1 | r} \end{matrix}]$ . Therefore, the augmented systems equations $Ψ_{r}$ and ${\tilde{Ψ}}_{r + 1}$ combining equations (1) and (9)–(11) are defined as:

Ψ_{r} = (A_{r 1} + H_{r 1} F_{r} E_{r 1}) {\tilde{Ψ}}_{r} + D_{r 1} v_{r},

(12)

{\tilde{Ψ}}_{r + 1} = (A_{r 2} + H_{r 2} F_{r} E_{r 2}) {\tilde{Ψ}}_{r} + B_{r 2} w_{r} + D_{r 2} v_{r} .

(13)

For the sake of brevity, the parameters are represented as

\begin{matrix} A_{r 1} = [\begin{matrix} I - K_{r} C_{r} K_{r} ({\hat{C}}_{r} - C_{r}) \\ K_{r} C_{r} I + K_{r} (C_{r} - {\hat{C}}_{r}) \end{matrix}], \\ H_{r 1} = [\begin{matrix} - K_{r} ℋ_{r} \\ K_{r} ℋ_{r} \end{matrix}], E_{r 1} = [E_{r} E_{r}], D_{r 1} = [\begin{matrix} - K_{r} \\ K_{r} \end{matrix}], \\ A_{r 2} = [\begin{matrix} A_{r} - L_{r} C_{r} A_{r} - {\hat{A}}_{r} + L_{r} ({\hat{C}}_{r} - C_{r}) \\ L_{r} C_{r} {\hat{A}}_{r} + L_{r} (C_{r} - {\hat{C}}_{r}) \end{matrix}], \\ H_{r 2} = [\begin{matrix} ℱ_{r} - L_{r} ℋ_{r} \\ L_{r} ℋ_{r} \end{matrix}], E_{r 2} = [E_{r} E_{r}], \\ B_{r 2} = [\begin{matrix} B_{r} \\ 0 \end{matrix}], D_{r 2} = [\begin{matrix} - L_{r} \\ L_{r} \end{matrix}] . \end{matrix}

(14)

The corresponding estimation error covariance matrices are denoted by ${\tilde{Θ}}_{r} = E (Ψ_{r} Ψ_{r}^{T})$ and ${\tilde{Σ}}_{r + 1} = E ({\tilde{Ψ}}_{r + 1} {\tilde{Ψ}}_{r + 1}^{T})$ based on vectors $Ψ_{r}$ and ${\tilde{Ψ}}_{r + 1}$ under equation (11), respectively. And then, according to Lemmas 2 and 3, the upper boundaries of the estimation error covariance matrices are defined by ${\bar{Θ}}_{r}$ and ${\bar{Σ}}_{r + 1}$ as follows:

E (e_{r} e_{r}^{T}) = [I 0] {\tilde{Θ}}_{r} [\begin{matrix} I \\ 0 \end{matrix}] ⩽ [I 0] Θ_{r} [\begin{matrix} I \\ 0 \end{matrix}] = {\bar{Θ}}_{r}

(15)

and

E ({\tilde{e}}_{r + 1} {\tilde{e}}_{r + 1}^{T}) = [I 0] {\tilde{Σ}}_{r + 1} [\begin{matrix} I \\ 0 \end{matrix}] ⩽ [I 0] Σ_{r + 1} [\begin{matrix} I \\ 0 \end{matrix}] = {\bar{Σ}}_{r + 1} .

(16)

Moreover, $Θ_{r}$ is derived from

\begin{matrix} {\tilde{Θ}}_{r} = E (Ψ_{r} Ψ_{r}^{T}) \\ = (A_{r 1} + H_{r 1} F_{r} E_{r 1}) {\tilde{Σ}}_{r} {(A_{r 1} + H_{r 1} F_{r} E_{r 1})}^{T} + D_{r 1} R_{r} D_{r 1}^{T} \\ \leq A_{r 1} {({\tilde{Σ}}_{r}^{- 1} - α_{r} E_{r 1}^{T} E_{r 1})}^{- 1} A_{r 1}^{T} + α_{r}^{- 1} H_{r 1} H_{r 1}^{T} + D_{r 1} R_{r} D_{r 1}^{T} \\ = A_{r 1} {\tilde{Σ}}_{r} A_{r 1}^{T} + α_{r}^{- 1} H_{r 1} H_{r 1}^{T} + D_{r 1} R_{r} D_{r 1}^{T} \\ + A_{r 1} {\tilde{Σ}}_{r} E_{r 1}^{T} (α_{r}^{- 1} I - E_{r 1} {\tilde{Σ}}_{r} E_{r 1}^{T}) E_{r 1} {\tilde{Σ}}_{r} A_{r 1}^{T} \\ \leq A_{r 1} Σ_{r} A_{r 1}^{T} + α_{r}^{- 1} H_{r 1} H_{r 1}^{T} + D_{r 1} R_{r} D_{r 1}^{T} \\ + A_{r 1} Σ_{r} E_{r 1}^{T} (α_{r}^{- 1} I - E_{r 1} {\tilde{Σ}}_{r} E_{r 1}^{T}) E_{r 1} Σ_{r} A_{r 1}^{T} \\ = Θ_{r} . \end{matrix}

(17)

Similarly, $Σ_{r + 1}$ is also obtained from the solution of the following recursive:

\begin{matrix} Σ_{r + 1} = A_{r 2} Σ_{r} A_{r 2}^{T} + α_{r}^{- 1} H_{r 2} H_{r 2}^{T} \\ + A_{r 2} Σ_{r} E_{r 2}^{T} (α_{r}^{- 1} I - E_{r 2} Σ_{r} E_{r 2}^{T})^{- 1} E_{r 2} Σ_{r} A_{r 2}^{T} \\ + B_{r 2} Q_{r} B_{r 2}^{T} + D_{r 2} R_{r} D_{r 2}^{T} \\ + B_{r 2} S_{r} D_{r 2}^{T} + D_{r 2} S_{r}^{T} B_{r 2}^{T} . \end{matrix}

(18)

Furthermore, the optimal values of the proposed finite horizon state estimation based on Kalman-type filtering in equations (9) and (10) are solved from Theorem 4.

Theorem 4. Set the current time instant as k, and the stored valid measurement $y_{k}$ with artificial transmission delay $τ'_{k}$ as well as the time-stamped measurement $z_{r}$ are defined. $r = k - τ'_{k}$ is received before being transmitted. Let $α_{r}$ be a positive scalar. ${\bar{Σ}}_{r}$ from equation (16) and state covariance matrix $P_{r}$ are the positive definite solutions for the following discrete-time Riccati-like iterations:

{\bar{Θ}}_{r} = {\bar{Σ}}_{r} + {\bar{Σ}}_{r} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} {\bar{Σ}}_{r} - Λ_{r} Ξ_{r}^{- 1} Λ_{r}^{T},

(19)

\begin{matrix} {\bar{Σ}}_{r + 1} = A_{r} {\bar{Σ}}_{r} (I + E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r}) A_{r}^{T} \\ - Δ_{r} Ξ_{r}^{- 1} Δ_{r}^{T} + B_{r} Q_{r} B_{r}^{T} + α_{r}^{- 1} F_{r} F_{r}^{T}, \end{matrix}

(20)

P_{r + 1} = A_{r} {(P_{r}^{- 1} - α_{r} E_{r}^{T} E_{r})}^{- 1} A_{r}^{T} + α_{r}^{- 1} F_{r} F_{r}^{T} + B_{r} Q_{r} B_{r}^{T},

(21)

where $Λ_{r} = {\bar{Σ}}_{r} C_{r}^{T} + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r} C_{r}^{T}$ and $Δ_{r} = A_{r} {\bar{Σ}}_{r} (I + E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r}) C_{r}^{T} + α_{r}^{- 1} ℱ_{r} ℋ_{r}^{T} + B_{r} S_{r}$ . Note that they satisfy $P_{r}^{- 1} - α_{r} E_{r}^{T} E_{r} > 0$ and $M_{r} = α_{r}^{- 1} I - E_{r} {\bar{Σ}}_{r} E_{r}^{T} > 0$ .

Then, the proposed state estimation approach based on the Kalman-type filtering given in equations (9) and (10) is able to solve the following filter parameters:

{\hat{C}}_{r} = C_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r}),

(22)

K_{r} = Λ_{r} Ξ_{r}^{- 1},

(23)

{\hat{A}}_{r} = A_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r}),

(24)

L_{r} = Δ_{r} Ξ_{r}^{- 1} .

(25)

Here, $Ξ_{r} = C_{r} {\bar{Σ}}_{r} (I + E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r}) C_{r}^{T} + α_{r}^{- 1} ℋ_{r} ℋ_{r}^{T} + R_{r}$ and ${\tilde{M}}_{r} = α_{r}^{- 1} I - E_{r} P_{r} E_{r}^{T}$ .

Proof. The proof procedure is the derivation on the basis of minimizing error covariance matrices.

Since the solutions of ${\bar{Θ}}_{r}$ and ${\bar{Σ}}_{r + 1}$ are derived from ${\bar{Σ}}_{r}$ defined in equations (15) and (16), the upper boundary matrix $Σ_{r}$ can be decomposed as follows (Rezaei et al. 2015):

Σ_{r} = [\begin{matrix} Σ_{11, r} Σ_{12, r} \\ Σ_{21, r} Σ_{22, r} \end{matrix}] = [\begin{array}{l} {\bar{Σ}}_{r} 0 \\ 0 P_{r} - {\bar{Σ}}_{r} \end{array}] .

(26)

Considering the given recursive equations for ${\bar{Σ}}_{r + 1}$ and $P_{r + 1}$ in equations (20) and (21), as well as the estimation ${\hat{x}}_{r | r}$ in equation (9), the estimated parameters ${\hat{C}}_{r}$ and $K_{r}$ are determined by minimizing measurement and estimation error covariance matrices, respectively.

Step 1: Calculate the filter parameter ${\hat{C}}_{r}$ .

First, the measurement error ${\tilde{y}}_{k}$ is determined by the stored valid data $z_{r}$ . According to the definition of ${\tilde{Ψ}}_{r}$ and $Ψ_{r}$ in equation (11), ${\tilde{y}}_{k}$ is defined as:

{\tilde{y}}_{k} = y_{k} - {\hat{y}}_{k} = (A_{r 3} + H_{r 3} F_{r} E_{r 3}) {\tilde{Ψ}}_{r} + v_{r},

(27)

where $A_{r 3} = [\begin{matrix} C_{r} & C_{r} - {\hat{C}}_{r} \end{matrix}]$ , $H_{r 3} = H_{r}$ , $E_{r 3} = [\begin{matrix} E_{r} & E_{r} \end{matrix}]$ and ${\hat{y}}_{k} = {\hat{C}}_{r} {\hat{x}}_{r | r - 1}$ is the estimated measurement.

Next, minimize the measurement error covariance matrix (i.e. the first-order derivative of $E ({\tilde{y}}_{k} {\tilde{y}}_{k}^{T})$ is set as $\frac{\partial Tr (E ({\tilde{y}}_{k} {\tilde{y}}_{k}^{T}))}{\partial {\hat{C}}_{r}} = 0$ , where $Tr (A A^{T})$ represents the trace of $A A^{T}$ ). Developing the results of Lemmas 2 and 3 and equation (26), solve the upper boundary of the measurement error covariance matrix, that is:

\begin{matrix} E ({\tilde{y}}_{k} {\tilde{y}}_{k}^{T}) \\ = (A_{r 3} + H_{r 3} F_{r} E_{r 3}) {\tilde{Σ}}_{r} (A_{r 3} + H_{r 3} F_{r} E_{r 3})^{T} + R_{r} \\ ⩽ A_{r 3} (Σ_{r} + Σ_{r} E_{r 3}^{T} (α_{r}^{- 1} I - E_{r 3} Σ_{r} E_{r 3}^{T})^{- 1} E_{r 3} Σ_{r}) A_{r 3}^{T} + α_{r}^{- 1} H_{r 3} H_{r 3}^{T} + R_{r} \\ = {\bar{Π}}_{r} . \end{matrix}

(28)

Substituting $A_{r 3}$ , $H_{r 3}$ , $E_{r 3}$ and $Σ_{r}$ into equation (28), ${\bar{Π}}_{r}$ is rewritten as

\begin{matrix} {\bar{Π}}_{r} = C_{r} {\bar{Σ}}_{r} C_{r}^{T} + (C_{r} - {\hat{C}}_{r}) (P_{r} - {\bar{Σ}}_{r}) {(C_{r} - {\hat{C}}_{r})}^{T} + α_{r}^{- 1} H_{r} H_{r}^{T} + R_{r} \\ + (C_{r} {\bar{Σ}}_{r} E_{r}^{T} + (C_{r} - {\hat{C}}_{r}) (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T}) {(α_{r}^{- 1} I - E_{r} P_{r} E_{r}^{T})}^{- 1} \\ \times {(C_{r} {\bar{Σ}}_{r} E_{r}^{T} + (C_{r} - {\hat{C}}_{r}) (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T})}^{T} . \end{matrix}

(29)

Therefore, using the first-order derivative from ${\bar{Π}}_{r}$ and ${\hat{C}}_{r}$ , the straightforward transformation of ${\hat{C}}_{r}$ from $\frac{\partial T r ({\bar{Π}}_{r})}{\partial {\hat{C}}_{r}} = 0$ is derived as follows:

{\hat{C}}_{r} = C_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} {(I + (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r})}^{- 1}),

(30)

where ${\tilde{M}}_{r} = α_{r}^{- 1} I - E_{r} P_{r} E_{r}^{T}$ , and define $M_{r} = α_{r}^{- 1} I - E_{r} {\bar{Σ}}_{r} E_{r}^{T}$ . Moreover, on the basis of the matrix inversion lemma, the replaced equivalent forms of $E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r}$ in equation (30) can be derived from:

\begin{matrix} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} \\ = E_{r}^{T} M_{r}^{- 1} E_{r} {(I + ({\bar{Σ}}_{r} - P_{r}) E_{r}^{T} M_{r}^{- 1} E_{r})}^{- 1} \\ = {(I + E_{r}^{T} M_{r}^{- 1} E_{r} ({\bar{Σ}}_{r} - P_{r}))}^{- 1} E_{r}^{T} M_{r}^{- 1} E_{r} . \end{matrix}

(31)

Furthermore, from the above analysis, substituting equation (31) into $I + (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r}$ given in equation (30), the equivalent form is

I + (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} = {(I + ({\bar{Σ}}_{r} - P_{r}) E_{r}^{T} M_{r}^{- 1} E_{r})}^{- 1} .

(32)

Therefore, the filter parameter ${\hat{C}}_{r}$ is determined as follows:

\begin{matrix} {\hat{C}}_{r} = C_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} {(I + (P_{r} - {\bar{Σ}}_{r}) E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r})}^{- 1}) \\ = C_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r} {(I + ({\bar{Σ}}_{r} - P_{r}) E_{r}^{T} M_{r}^{- 1} E_{r})}^{- 1} \\ \times (I + ({\bar{Σ}}_{r} - P_{r}) E_{r}^{T} M_{r}^{- 1} E_{r})) \\ = C_{r} (I + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r}) . \end{matrix}

(33)

Step 2: Obtain the filter gain $K_{r}$ .

First, the solutions of $Θ_{r}$ and $Σ_{r + 1}$ are defined in equations (17) and (18), respectively. From equations (15) and (16) and the solved ${\hat{C}}_{r}$ in Step 1, the corresponding filtering and prediction error covariance matrices ${\bar{Θ}}_{r}$ and ${\bar{Σ}}_{r + 1}$ are obtained as follows:

\begin{matrix} {\bar{Θ}}_{r} = [\begin{matrix} I & 0 \end{matrix}] Θ_{r} [\begin{matrix} I \\ 0 \end{matrix}] \\ = (I - K_{r} C_{r}) {\bar{Σ}}_{r} {(I - K_{r} C_{r})}^{T} \\ + (K_{r} ({\hat{C}}_{r} - C_{r})) (P_{r} - {\bar{Σ}}_{r}) {(K_{r} ({\hat{C}}_{r} - C_{r}))}^{T} \\ + ({\bar{Σ}}_{r} + K_{r} ({\hat{C}}_{r} (P_{r} - {\bar{Σ}}_{r}) - C_{r} P_{r})) E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} \\ \times {({\bar{Σ}}_{r} + K_{r} ({\hat{C}}_{r} (P_{r} - {\bar{Σ}}_{r}) - C_{r} P_{r}))}^{T} \\ + α_{r}^{- 1} K_{r} H_{r} H_{r}^{T} K_{r}^{T} + K_{r} R_{r} K_{r}^{T} . \end{matrix}

(34)

Note that $K_{r}$ is obtained depending on ${\hat{C}}_{r}$ and equation (34). After deduction of minimizing error covariance and the first-order derivative from $\frac{\partial Tr ({\bar{Θ}}_{r})}{\partial {\hat{K}}_{r}} = 0$ , $K_{r}$ is derived as follows:

K_{r} = Λ_{r} Ξ_{r}^{- 1},

(35)

in which

\begin{matrix} Λ_{r} = {\bar{Σ}}_{r} C_{r}^{T} + {\bar{Σ}}_{r} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} {(C_{r} P_{r} - {\hat{C}}_{r} (P_{r} - {\bar{Σ}}_{r}))}^{T} \\ = {\bar{Σ}}_{r} C_{r}^{T} + {\bar{Σ}}_{r} E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r} C_{r}^{T}, \end{matrix}

(36)

and

Ξ_{r} = C_{r} {\bar{Σ}}_{r} (I + E_{r}^{T} M_{r}^{- 1} E_{r} {\bar{Σ}}_{r}) C_{r}^{T} + α_{r}^{- 1} H_{r} H_{r}^{T} + R_{r} .

(37)

Step 3: Solve the filter parameters ${\hat{A}}_{r}$ and $L_{r}$ .

Similar to the computational procedure of ${\hat{C}}_{r}$ and $K_{r}$ , filter parameters ${\hat{A}}_{r}$ and $L_{r}$ are solved from ${\bar{Σ}}_{r + 1}$ in equation (16). Then, the iteration forms are obtained from equations (24) and (25), respectively.

Step 4: Calculate the covariance matrix $P_{r + 1}$ .

Depending on the results of Lemmas 2 and 3, the upper boundary for the state covariance matrix is obtained as

\begin{matrix} {\tilde{P}}_{r + 1} = E (x_{r + 1} x_{r + 1}^{T}) \\ = (A_{r} + F_{r} F_{r} E_{r}) P_{r} {(A_{r} + F_{r} F_{r} E_{r})}^{T} + B_{r} Q_{r} B_{r}^{T} \\ ⩽ A_{r} {(P_{r}^{- 1} - α_{r} E_{r}^{T} E_{r})}^{- 1} A_{r}^{T} + α_{r}^{- 1} F_{r} F_{r}^{T} + B_{r} Q_{r} B_{r}^{T} \\ \begin{matrix} = A_{r} P_{r} A_{r}^{T} + A_{r} P_{r} E_{r}^{T} {\tilde{M}}_{r}^{- 1} E_{r} P_{r} A_{r}^{T} + α_{r}^{- 1} F_{r} F_{r}^{T} + B_{r} Q_{r} B_{r}^{T} \\ = P_{r + 1} . \end{matrix} \end{matrix}

(38)

Note that the initial value is set as $P_{0} = x_{0} x_{0}^{T} + P_{0}$ , and a similar calculation is investigated by Feng et al. (2013). Therefore, $P_{r + 1}$ is obtained in equation (21).

Estimation-based delay compensation

The event-driven signal-choosing scheme is able to drop the out-of-order packets. However, the LZOH inevitably produces the missing data packets. Therefore, the obtained optimal state estimation ${\hat{x}}_{r | r}$ is used for compensating the missing state estimation of ${{\hat{x}}_{r + 1 | r}, \dots, {\hat{x}}_{k | r}}$ at the time instant of k.

To reduce the computational burden with transmission delay, an estimation-based compensation method for random delays is proposed, which is represented as follows:

{\hat{x}}_{k | r} = (1 - \frac{τ'_{k} - 1}{N}) {\hat{x}}_{r + 1 | r} .

(39)

First, set the largest transmission delay to N, and the acknowledged data packet is $y_{k}$ . Then, it is assumed that the received valid data packet is $z_{r}$ with transmission delay $τ'_{k}$ at the current time instant. Furthermore, the state estimated value ${\hat{x}}_{r | r}$ is used for compensating the estimation-based state ${\hat{x}}_{k | r}$ .

Note that the proposed estimation-based compensation method is an approximate delay-free linear state estimation. In accordance with the role of the LZOH scheme, the estimated state for the latest data packet ${\hat{x}}_{r | r}$ is stored with a timestamp that is closest to the actual state. If the transmission delay is larger ( $τ'_{k} \leq N$ ), the estimation accuracy is smaller, which is dependent on the prediction of ${\hat{x}}_{r + 1 | r}$ . Therefore, the state estimation with random transmission delay is obtained from the predicted state ${\hat{x}}_{r + 1 | r}$ .

Remark 5. The issue of state estimation with transmission delays is interesting in the compensation strategy of one-step prediction, which is an accurate estimation due to using the current estimation. However, the computational complexity is increased, and a lot of energy resources will be consumed. Since the proposed estimation-based compensation strategy is an approximate state estimation, the estimation accuracy of the proposed strategy is lower than one-step prediction compensation. It is worth noticing that the proposed strategy possesses the significant advantage of suppressing the computational burden and improving the estimation efficiency.

The stored data are $y_{k + 1}$ with delay $τ'_{k + 1}$ , and the received measurement is $z_{s}$ with time-stamp before being transmitted, at the sampling time $k + 1$ . Set $s = k + 1 - τ'_{k + 1}$ , and satisfy $s ⩾ r$ . There are two cases for obtaining the estimated state ${\hat{x}}_{s | s}$ :

Case 1 For $s = r$ or $s = r + 1$ , the state estimation ${\hat{x}}_{s | s}$ is derived from equation (9), and the filter parameters are obtained by the recursive equations based on Theorem 4.

Case 2 If $s > r + 1$ , the estimated state ${\hat{x}}_{s | s}$ will be compensated by one-step prediction of ${\hat{x}}_{r + 1 | r + 1}$ , which is calculated from equations (9)–(10). Therefore, develop the artificial delay satisfying $τ_{k}^{sr} = s - r > 1$ , and the reorganized compensated sequence is defined as:

{{\hat{x}}_{r + 1 | r + 1}, \dots, {\hat{x}}_{s | s}} .

(40)

Then, for the given systems (1) and (2), the missing states are compensated and the filter parameters are computed by the recursions in equations (19)–(25).

It is worth mentioning that due to the role of the LZOH scheme under a stochastic system, the case of $s < r$ named out-of-order packets is non-existent. In this sense, the LZOH is more reasonable for improving system performance and reducing computational complexity. Algorithm 1 summarizes the event-based finite horizon state estimation using the LZOH scheme.

Algorithm 1 Computation procedure of event-based finite horizon state estimation
Input: Initial state $x_{0}$ , estimated state ${\hat{x}}_{0 \| 0} = μ_{0}$ with covariance $P_{0}$ ,prediction error covariance ${\bar{Σ}}_{0}$ , positive scalar $α_{0}$ and sampling time t.
Output: Estimation error covariance ${\bar{Σ}}_{k + 1}$ and $P_{k + 1}$ .
for $k = 1 \to t$ do
$r = k - τ'_{k}$ ; /* $τ'_{k}$ -step delays */
if $τ_{k}^{sr} = 0$ or $τ_{k}^{sr} = 1$ then
${\bar{Θ}}_{r}$ in equation (19);
/* the upper boundary of estimation error covariance substituted from ${\bar{Σ}}_{r}$ */
${\hat{C}}_{r}$ , $K_{r}$ , ${\hat{A}}_{r}$ and $L_{r}$ given in equations (22)–(25);
/* the filter parameters derived from ${\bar{Σ}}_{r}$ */
${\hat{x}}_{r \| r}$ from equation (9), and ${\hat{x}}_{r + 1 \| r}$ from equation (10);
/* design the filter and predictor */
${\bar{Σ}}_{r + 1}$ in equation (20), and $P_{r + 1}$ in equation (21);
/* the estimation error covariance and state covariance */
else
If $τ_{k}^{sr} > 1$ then
${\hat{x}}_{s \| s}$ is compensated by ${{\hat{x}}_{r + 1 \| r + 1}, \dots, {\hat{x}}_{s \| s}}$ calculated from equations (9) and (10);
end if
end if
${\hat{x}}_{k \| r}$ under equation (39); /* the state of estimation-based compensation for delays */
$s = k + 1 - τ'_{k + 1}$ ;
/* $τ'_{k + 1}$ -step delays at the following sampling time instant */
$τ_{k}^{sr} = s - r$ ; /* the artificial delay */
end for

Numerical simulation

A numerical example is provided to verify the effectiveness of the proposed finite horizon state estimation method.

The considered stochastic systems are target-tracking systems with intermittent measurements in the work of Feng et al. (2013), Tian et al. (2016), Wang et al. (2015) and Rezaei et al. (2015):

\begin{matrix} x_{k + 1} = ([\begin{matrix} 0.9 T T^{2} / 2 \\ 0 0.9 T \\ 0 0 0.9 \end{matrix}] + F_{k} F_{k} E_{k}) x_{k} + [\begin{matrix} T^{2} / 2 \\ T \\ 1 \end{matrix}] w_{k}, \\ k = 1, 2, \dots \end{matrix}

(41)

z_{k} = (C_{k} + H_{k} F_{k} E_{k}) x_{k} + v_{k},

(42)

w_{k} = η_{k},

(43)

v_{k} = ζ w_{k} .

(44)

Set $T = 0.1$ s as the sampling period. Suppose the maximum of the transmission delay is $N = 5$ , and the time-varying parametric uncertainties satisfy $F_{k} = \sin (0.6 k)$ . The estimated state $x_{k} = {(s_{k} {\overset{\cdot}{s}}_{k} {\overset{\cdot\cdot}{s}}_{k})}^{T}$ is composed of the position, velocity and acceleration, respectively. For noise, $η_{k} \in R$ is the zero-mean white noise with variance $σ_{η}^{2} = 0.09$ , and the correlated strength of the variable $ζ$ in equation (43) is determined as $ζ = 2$ . Set $F_{k} = {[\begin{matrix} 0.1 & 0.1 & 0.1 \end{matrix}]}^{T}$ , $E_{k} = [\begin{matrix} 0.02 & 0.02 & 0.02 \end{matrix}]$ , $C_{k} = [\begin{matrix} 0.6 & 0.8 & 1 \end{matrix}]$ and $H_{k} = 0.8$ . Taking into account the correlation between process noise and measurement noise, the process noise $w_{k}$ from equation (43) with unity covariance $Q_{k}$ , and the covariance $R_{k}$ is denoted as $R_{k} = ζ Q_{k} ζ^{T}$ , as well as $S_{k} = Q_{k} ζ^{T}$ given in equation (44).

Note that the initial values are denoted as ${\hat{x}}_{0 | 0} = μ_{0} = E (x_{0}) = {[\begin{matrix} 1 & 1 & 1 \end{matrix}]}^{T}$ and $P_{0 | 0} = 0.01 I_{3}$ . The traces of the estimation error covariance matrices are shown in Figure 3(a), and the comparison of the delay-free estimated states ${\hat{x}}_{k | k}$ are demonstrated in Figures 3(b)–(d), using the proposed method and the improved robust finite horizon Kalman filtering (IRFHKF) (Rezaei et al. 2015) at the time instant of k. Figure 3 implies that the estimation error covariance upper boundary obtained from the proposed approach is remarkably less than the IRFHKF. Owing to designing the appropriate estimator of the proposed method by probing the optimal filter parameters, and considering the correlation of noise, the dynamic tracking trajectory is closer to the actual state at the current time instant.

To further illustrate the effectiveness performance, the corresponding tracking results of the estimated states are shown in Figure 4. The simulation results are obtained from equations (9) and (10), which are calculated in Theorem 4 for the event-driven LZOH. The comparison results in Figure 4 include the proposed estimation method, estimated state with out-of-order packets (ZOH) and IRFHKF method.

Figure 4 shows the actual state and the estimated state with holding or dropping out-of-order packets. It is observed that the LZOH scheme is able to discard out-of-order packets induced from random transmission delays. Furthermore, using the LZOH scheme provides better performance for target tracking and improved computational efficiency in the compensation strategy.

In order to demonstrate the performance of the proposed finite horizon state estimation with transmission delays and out-of-order packets, the mean-square error (MSE) values (Chen et al. 2014a; Tian et al. 2016; Zhang et al. 2015) used for solving the mean-square of the estimation error values, as well as the running time of each method, are presented in Table 1.

Table 1.

Comparison for mean-square errors and running times.

Scheme	Mean of error covariance			Time
Scheme	Position	Velocity	Acceleration	Time
Non-event-driven mechanism	0.0063	0.0095	0.0026	0.2340
LZOH scheme	0.0042	0.0079	0.0022	0.1248
IRFHKF	0.0124	0.0148	0.0322	1.8564

Table 1 includes the estimated states of position, velocity and acceleration according to the LZOH scheme, non-event-driven mechanism and IRFHKF method. It shows that the MSEs for LZOH are smaller than for the non-event-driven mechanism and IRFHKF. Meanwhile, the running time is shorter than the other methods. Therefore, choosing the latest data packet allows more accurate estimation performance than the estimated state with out-of-order packets and IRFHKF method.

Depending on the simulation and error covariance criteria, the comparison between the proposed method and IRFHKF implies that the proposed estimator for event-driven mechanism has better accuracy than IRFHKF for the multi-step random delays. The actual estimation error covariance is below the upper boundaries, so the proposed estimator with transmission delays and out-of-order packets shows the superiority in performance for following the actual state.

Conclusion

With the aid of a linear delay compensation strategy, the optimal state estimation issue based on event-driven finite horizon state estimation is investigated. To analyse the networked-induced phenomena and reduce energy consumption, the system is modelled by the event-driven LZOH scheme, which deals with packet disorders by sequence reorganization. Based on the established model, a finite horizon state estimation is probed by the solution of a Riccati-like matrix recursive equation guaranteeing the optimized upper boundary. Moreover, the estimation-based compensation for the random transmission delays is proposed to improve the estimator performance and reduce the computational complexity. As a theoretical result for the proposed event-based estimation approach, the LZOH scheme possesses superior performance for dropping packet disorders, and the actual estimation error variances for the states is lower than their upper boundaries. Moreover, the measurement accuracy obtained from the processor is improved. Target-tracking system and numerical simulations are performed to demonstrate that the estimator has the ability to track the actual state.

Footnotes

Acknowledgements

The authors would like to thank the editors and anonymous reviewers for their valuable comments and helpful suggestions, which greatly improved the quality of this paper.

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

This work was supported by the Natural Science Foundation of China (Grant nos. 61633016, 61472172 and 61772253), Key Project of Science and Technology Commission of Shanghai Municipality (Grant nos. 15220710400 and 15411953502), Natural Science Foundation of Shanghai (Grant no. 18ZR1415100), Key Technical of Key Industries of Shandong (Grant no. 2016CYJS03A02-1) and Key Research and Development Project of Shandong (Grant no. 2016ZDJS06A05).

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