Event-triggered L 1 filtering for uncertain networked control systems with multiple sensor fault modes

Abstract

Considering the influence of sensor fault modes on the system performance in communication network, under the discrete event-triggered communication scheme (DETCS), the problems of the robust $L_{1}$ filtering for a class of uncertain networked control systems (NCSs) with multiple sensor fault modes and persistent and amplitude-bounded disturbance constraints are investigated. A set of stochastic variables are adopted to describe the sensor faults, the filtering error system is established, which characterizes the effects of sensor faults and DETCS. By constructing an appropriate delay-dependent Lyapunov-Krasovskii functional, new results on stability and robust $L_{1}$ performance are proposed for the filtering error system according to Lyapunov theory and the integral inequality method, and the co-design method for gaining the desired $L_{1}$ filter parameters and event-triggering parameters is given in terms of linear matrix inequalities (LMIs). We notice that with the structrue of Lyapunov-Krasovskii functional which considers the piecewise-linear sawtooth structure characteristic of transmission delay, a less conservative result is obtained. Finally, three examples are provided to illustrate the feasibility of the proposed method.

Keywords

Co-design discrete event-triggered communication scheme (DETCS)sensor fault modes networked control systems (NCSs)

Introduction

In the past few decades, fruitful research results of robust filtering have been reported due to the fact for many practical filtering applications; see, for example, Wu et al. (2017); Boada and Boada (2018); Wang et al. (2019c); and Zanni et al. (2016) and the references therein. In particular, with the in-depth study of robust filtering analysis and synthesis, some robust filter design methods are proposed by introducing different norms (such as $H_{\infty}, L_{2}$ - $L_{\infty}$ , Hankel). These norms are limited by an implicit assumption that the external disturbances must be energy-bounded. However, some actual systems may be subjected to persistent heavy noise interference, for instance, aircraft flight control systems, shipping navigation control systems, and temperature control systems of heating furnace, under the circumstances, the $L_{1}$ norm appears to be more suitable as a measurement index of input and output signals (Chen et al., 2017; Liu and Xiang, 2014; Qi et al., 2018b; Wang et al., 2019a). Different performance constraints determine different analytical skills. Relatively speaking, there are few reports about $L_{1}$ filter design methods (Mohammadian et al., 2015; Li et al., 2016b; Qi et al., 2018a; Qi et al., 2019), and in most cases, it is difficult to solve $L_{1}$ filter analytically.

With the rapid development of engineering applications of remote sensing robot, intelligent transportation and chemical industry, an ever-increasing research attention has been attracting on networked control systems (NCSs) (Li and Chen, 2018; Qiu et al., 2016; Zhang et al., 2017a). However, the constraints of network bandwidth, data collision, queuing and connection interruption, which will be delays and packet dropouts in the process of information transmission. Consequently, various reliable control and filtering approaches have been developed to solve these challenging problems; see, for example, Kargar et al. (2018) and Wang et al. (2013) and the references therein. In addition, the sensors in the communication network are prone to aging and zero drift, which will inevitably lead to sensor faults and affect system performance. Probabilistic sensor faults are more representative and should be considered in the filter design procedure (Li et al., 2016a; Tian and Yue, 2011). Also, for all we know, the $L_{1}$ filter design problem for uncertain NCSs with randomly occurring sensor faults and persistent and amplitude-bounded disturbance constraints has not been investigated, and the aim of this paper is to fill the gaps in the field.

In most of the existing literature, control or filtering problem for NCSs is implemental by utilizing a periodic time-triggered communication scheme (PTTCS). The advantages of PTTCS are that the well-developed theory on sampled-data control systems maybe used in the analysis and synthesis of NCSs, and easy implementation and good predictability. However, the PTTCS becomes less preferable because it may lead to unnecessary data transmissions, which will waste communication resources. As such, the event-triggered communication scheme appears with hope to improve the communication utilization and energy saving. Such a communication scheme provides an effective way of determining whether the current sampled data by sensors should be sent or not, which ensures that only “necessary data” can be sent over the network. So far, the control or filtering issue has gained a great deal of research attention for NCSs based on the event-triggered communication scheme (Dong et al., 2016; Watt, 2012; Peng and Fei, 2013; Ning et al., 2017). For example, in Watt, 2012 by using a time-delay approach, the filtering error system is modeled as a switched system under event-triggered scheme and the $H_{\infty}$ filtering problem is concerned. Also, to reduce information transmission, the $H_{\infty}$ state estimation problem is investigated for a class of uncertain discrete-time neural networks subject to infinitely distributed delays and fading channels in Dong et al. (2016). Although significant progress has been made in the study of event-triggered control or filtering or state estimation of NCSs in references, to the best of authors’ knowledge, there is no published research on co-design of the discrete event-triggered communication scheme (DETCS) and the $L_{1}$ filter in a unified framework for uncertain NCSs with persistent heavy noise disturbances, which motivates the current study.

Driven by the recent developments in NCSs and the increasing interest in DETCS, we investigate the problem of event-triggered robust $L_{1}$ filtering for uncertain NCSs with sensor faults. The main contributions of this paper can be summarized as follows:

To improve the efficiency in communication resource utilization and save energy, the DETCS is employed in this paper. Compared with continuous event-triggered communication, the extra hardware is not necessary since we only monitor the state and compute error at discrete instants, which will be implemented easily.

A novel Lyapunov-Krasovskii functional with the piecewise-linear sawtooth structure characteristic of transmission delay is adopted, which contributes to reduce the design conservatism. It should be stressed in particular that we have encountered difficulties in establishing $L_{1}$ performance criterion, in contrast to our previous research (Li et al., 2016b), the $L_{1}$ norm is more complex due to the fact that we adopt piecewise Lyapunov-Krasovskii functional while considering the DETCS, the $L_{1}$ performance criterion is constructed finally by flexibly using of integral inequalities and combining with Lyapunov-Krasovskii functional and its derivative terms. As a result, the sufficient condition is obtained for the existence of $L_{1}$ performance criterion that guarantees mean-square asymptotic stability and satisfies the worst-case peak-to-peak performance constraint of the filtering error system, and furthermore, the co-design scheme of the $L_{1}$ filtering and the DETCS is solved by means of linear matrix inequalities (LMIs).

The sensors in the communication network are prone to aging, zero drift and electro-magnetic interference, which will inevitably lead to sensor faults and affect system performance. The design method in this paper has certain tolerance for the stochastic sensor faults. In addition, it is worth mentioning that our main results are also valid for the unmanned surface vehicles (USVs) in the network environment.

Problem formulation

Consider a typical NCS with parameter uncertainty as follows

Σ : {\begin{cases} \overset{\cdot}{x} (t) = (A + Δ A) x (t) + B ω (t), \\ y (t) = C x (t), \\ z (t) = L x (t) \end{cases}

(1)

where $x (t) \in ℜ^{n}$ is the state vector, $y (t) \in ℜ^{l}$ is the measured output vector, $z (t) \in ℜ^{q}$ is the signal to be estimated, $ω (t) \in ℜ^{m}$ is the exogenous disturbance input signal with $ω (t)$ that belongs to $L_{\infty} [0, \infty)$ , $A, B, C, L$ are constant matrices of compatible dimensions and $Δ A$ is unknown parameter matrix that satisfy $Δ A = HF (t) E$ , where $H, E$ are known real constant matrices of compatible dimensions, and $F (t)$ is a real uncertain matrix function with Lebesgue measurable elements satisfying $F^{T} (t) F (t) \leq I$ .

Assuming that the sensor with probabilistic faults, then measurable output signal $y (t)$ of the system can be rewritten as

\hat{y} (t) = Ξ Cx (t)

(2)

where $Ξ = diag {Ξ_{1}, Ξ_{2}, \dots, Ξ_{k}}$ are the fault matrices with $Ξ_{i} (i = 1, 2, \dots, k)$ being k unrelated random variables taking values on the interval $[0, θ], θ \geq 1$ . Suppose the mathematical expectation and variance of $Ξ_{i}$ are ${\bar{Ξ}}_{i}$ and $δ_{i}^{2}$ , respectively, the ${\bar{Ξ}}_{i}$ and $δ_{i}^{2}$ can determined the failure rate and the distortion degree of the i th sensor. $Ξ_{i} = 1$ denotes that the i th sensor works normally, $Ξ_{i} = 0$ indicates that the i th sensor is invalid completely, and $0 < Ξ_{i} < 1$ and $Ξ_{i} > 1$ imply that the i th sensor is partial failure, that is, the disordering happens.

Define $E_{l} = diag {\underset{l - 1}{\underset{︸}{0, \dots, 0}}, 1, \underset{k - 1}{\underset{︸}{0, \dots, 0}}}$ , $\bar{Ξ} = diag {{\bar{Ξ}}_{1}, \dots, {\bar{Ξ}}_{k}}$ , then $\bar{Ξ} = \sum_{l = 1}^{k} {\bar{Ξ}}_{l} E_{l}$ . For any matrix $Θ > 0$ , one has

{\begin{matrix} E {Ξ - \bar{Ξ}} = 0, \\ E {{(Ξ - \bar{Ξ})}^{T} Θ (Ξ - \bar{Ξ})} = \sum_{l = 1}^{k} δ_{i}^{2} E_{l}^{T} Θ E_{l} \end{matrix}

(3)

Considering the filter for the estimation of $z (t)$ as follows

Σ_{f} : {\begin{matrix} {\overset{\cdot}{x}}_{f} (t) = A_{f} x_{f} (t) + B_{f} \hat{y} (t), \\ z_{f} (t) = C_{f} x_{f} (t) \end{matrix}

(4)

where $A_{f}, B_{f}, C_{f}$ are the filter parameters to be designed, $x_{f} (t), z_{f} (t)$ are the state vector and output vector of the filter, respectively, $\hat{y} (t)$ is the sensor output signal.

For the purpose of solving the waste of network bandwidth resource problem in PTTCS, the DETCS is introduced to provide an effective way of determining whether the current sampled data by sensors should be sent out to the filter or not, as shown in Figure 1 (Gu et al., 2016).

Figure 1.

Structure diagram of event-triggered $L_{1}$ filtering for NCSs with multiple sensor faults.

For the convenience of theoretical development, we make the following assumptions, which are common in NCSs research reported in the open literatures (Chen and Han, 2013; Zhang et al., 2017b).

Assumption 1: The transmission event is determine by a DETCS and the system’s state is periodically sampled at a constant sampling period $h > 0$ . At instant $t_{k} h$ , the sending period $h_{k} = t_{k + 1} h - t_{k} h$ , where $k \in ℜ^{+}, t_{k} \in ℜ^{+}$ and $li m_{k \to \infty} t_{k} \to \infty$ . At the same time, the sampling sequence is described by the set $X_{1} = {0, h, 2 h, \cdot \cdot \cdot, kh}$ , the transmission sequence is described by the set $X_{2} = {0, t_{1} h, \cdot \cdot \cdot, t_{k} h} \subseteq X_{1}$ .

Assumption 2: The network synthesis delay at instant $t_{k} h$ is $τ_{t_{k}}$ , and $τ_{t_{k}}$ denotes the transmission time from the instant $t_{k} h$ to the instant when the data has reached the filter. Therefore, the sampled sensor measurements $(y (t_{0} h), y (t_{1} h), y (t_{2} h), \cdot \cdot \cdot)$ arrive at the filter at instant $(t_{0} h + τ_{t_{0}}, t_{1} h + τ_{t_{1}}, t_{2} h + τ_{t_{2}}, \cdot \cdot \cdot)$ .

Assumption 3: The measurement output $\hat{y} (t)$ is realized through a zero-order hold (ZOH) with holding time $t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$ .

According to Assumption 2 and Assumption 3, $\hat{y} (t)$ in equation (2) can be described as

\hat{y} (t) = \hat{y} (t_{k} h) = Ξ Cx (t_{k} h) = \sum_{l = 1}^{k} Ξ_{l} E_{l} Cx (t_{k} h), t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})

(5)

We adopt the following discrete event-triggered condition

\begin{matrix} [E {\bar{Ξ} \hat{y} (i_{k} h) - \bar{Ξ} \hat{y} (t_{k} h)}]^{T} Φ [E {\bar{Ξ} \hat{y} (i_{k} h) - \bar{Ξ} \hat{y} (t_{k} h)}] \\ \leq σ E [{\bar{Ξ} \hat{y} (i_{k} h)}]^{T} Φ [E {\bar{Ξ} \hat{y} (i_{k} h)}] \end{matrix}

(6)

where $\hat{y} (i_{k} h)$ and $\hat{y} (t_{k} h)$ are the current sampled sensor measurements and the latest sensor measurements satisfying the trigger condition, respectively, and $i_{k} h = t_{k} h + lh (l = 1, 2, \cdot \cdot \cdot)$ . $Φ > 0$ is the weighting matrix, $σ > 0$ is the threshold, the parameters $Φ$ and $σ$ determine how frequently the sampled-data should be transmitted.

To determine whether the current sampled-data should be transmitted, an effective method is to consider the sampled-data error at every sampling instant, thus, the holding interval of the ZOH can be divided

Ω = [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}}) = Δ_{k}^{0} \cup Δ_{k}^{1} \cup \cdot \cdot \cdot \cup Δ_{k}^{d_{k}}

(7)

where $Δ_{k}^{l_{k}} = [i_{k} h + τ_{i_{k} h}, i_{k} h + h + τ_{i_{k} h + h})$ , $i_{k} h = t_{k} h + l_{k} h (l_{k} = 0, 1, 2, \cdot \cdot \cdot, d_{k}, d_{k} = t_{k + 1} - t_{k} - 1)$ .

Define

τ (t) = t - i_{k} h, (t \in Δ_{k}^{l_{k}})

(8)

It is clear that $τ (t)$ is a piecewise linear function satisfying

\overset{\cdot}{τ} (t) = 1, τ_{1} < τ_{i_{k} h} \leq τ (t) \leq h + τ_{i_{k} h + h} \leq h + \bar{τ} = τ_{2}

(9)

where $τ_{1} = min {τ_{l_{k}}}, \bar{τ} = max {τ_{t_{k}}}$ .

Define the mathematical expectation of the sensor measurement error between the current sampling instant and the latest transmission instant

E {\bar{Ξ} e_{k} (t)} = E {\bar{Ξ} \hat{y} (i_{k} h) - \bar{Ξ} \hat{y} (t_{k} h)} = E {\bar{Ξ} \hat{y} (t - τ (t)) - \bar{Ξ} \hat{y} (t_{k} h)}

(10)

Furthermore, based on equation (5), sampling technique and ZOH, the actual measurable output signal of the sensor can be described as

\hat{y} (t) = \bar{Ξ} Cx (t - τ (t)) + (Ξ - \bar{Ξ}) Cx (t - τ (t)) - \bar{Ξ} e_{k} (t)

(11)

Then, the filter $Σ_{f}$ in (4) is given finally by

Σ_{f} : {\begin{matrix} {\overset{\cdot}{x}}_{f} (t) = A_{f} x_{f} (t) + B_{f} [\bar{Ξ} Cx (t - τ (t)) + (Ξ - \bar{Ξ}) Cx (t - τ (t)) - \bar{Ξ} e_{k} (t)], \\ z_{f} (t) = C_{f} x_{f} (t) \end{matrix}

(12)

Denote $ξ (t) = [x^{T} (t) x_{f}^{T} (t)]^{T}, \tilde{z} (t) = z (t) - z_{f} (t)$ , augmenting the system $Σ$ in (1) to include the states of the filter $Σ_{f}$ in (12), we obtain the following filtering error system

\bar{Σ} : {\begin{matrix} \overset{\cdot}{ξ} (t) = \bar{A} ξ (t) + {\bar{A}}_{1} G ξ (t - τ (t)) + {\bar{A}}_{2} G e_{k} (t) + \bar{B} ω (t), \\ \tilde{z} (t) = \bar{C} ξ (t) \end{matrix}

(13)

where

{\bar{A}}_{1} = {\bar{A}}_{11} + {\bar{A}}_{12}, {\bar{A}}_{2} = {\bar{A}}_{21} + {\bar{A}}_{22}, \bar{C} = [\begin{matrix} L & - C_{f} \end{matrix}], G = [\begin{matrix} I & 0 \end{matrix}],

\bar{A} = [\begin{matrix} A + Δ A & 0 \\ 0 & A_{f} \end{matrix}], {\bar{A}}_{11} = [\begin{matrix} 0 \\ B_{f} \bar{Ξ} C \end{matrix}], {\bar{A}}_{12} = [\begin{matrix} 0 \\ B_{f} (Ξ - \bar{Ξ}) C \end{matrix}],

{\bar{A}}_{21} = [\begin{matrix} 0 \\ - B_{f} \bar{Ξ} \end{matrix}], {\bar{A}}_{22} = [\begin{matrix} 0 \\ - B_{f} (Ξ - \bar{Ξ}) \end{matrix}], \bar{B} = [\begin{matrix} B \\ 0 \end{matrix}] .

Remark 1: The DETCS and the multiple sensor fault modes are proposed, then an overall network-based filtering model with the consideration of DETCS and sensor faults is developed. Compared with continuous event-triggered communication in Li et al. (2015), the DETCS employed in this paper does not require the extra hardware since we only compute the sensor measurement error at a constant sampling period, which will be implemented easily. In addition, the threshold $σ$ in the inequality (6) is an adjustable parameter to achieve the satisfactory filtering performance, particularly, when $σ = 0$ , all the sampled-data should be transmitted, which means that the DETCS proposed in this paper reduces to the PTTCS.

The research objective of this paper can be expressed as follows. For the NCS $Σ$ in (1), we design a filter in the form of $Σ_{f}$ in (4) such that the filtering error system $\bar{Σ}$ in (13) is mean-square asymptotically stable with an $L_{1}$ noise attenuation level $γ > 0$ if it is mean-square asymptotically stable when $ω (t) = 0$ , and under zero initial condition and for all nonzero $ω (t) \in L_{\infty} [0, \infty)$ , the following holds

∥ T_{\tilde{z} ω} ∥_{L_{1}} = sup_{ω (t) \in L_{\infty} [0, \infty)} \frac{E {∥ \tilde{z} (t) ∥}_{L_{\infty}}}{∥ ω (t) ∥_{L_{\infty}}} < γ

Thus, $∥ T_{\tilde{z} ω} ∥_{L_{1}}$ corresponds to the maximum peak-to-peak gain of $T_{\tilde{z} ω}$ .

Before ending this subsection, we introduce the following lemma that will play key roles in our derivation.

Lemma 1: (Mathiyalagan et al., 2014): Given appropriately dimensioned matrices $ϒ, ϒ_{1}, ϒ_{2}$ , with $ϒ = ϒ^{T}$ then

ϒ + ϒ_{1} F (t) ϒ_{2} + (ϒ_{1} F (t) ϒ_{2})^{T} < 0

holds for all $F (t)$ satisfying $F (t) (F (t))^{T} \leq I$ if and only if for some $ϵ > 0$

ϒ + ϵ ϒ_{1}^{T} ϒ_{1} + ϵ^{- 1} ϒ_{2}^{T} ϒ_{2} < 0

Main results

$L_{1}$ performance analysis under DETCS

In this subsection, we give complete proofs of $L_{1}$ performance criterion for the filtering error system $\bar{Σ}$ in (13) with transmission delay and sensor faults, the following theorem plays an important role in the co-design of the robust $L_{1}$ filter and the DETCS.

Theorem 1: Given positive scalars $α, τ_{1}, τ_{2}, η, ν, σ, δ$ , under the DETCS, the filtering error system $\bar{Σ}$ in (13) is mean-square asymptotically stable with an $L_{1}$ performance level $γ$ if there exist matrices $P > 0$ , $Φ > 0$ , $Q_{i} > 0 (i = 1, 2, 3)$ , $R_{j} > 0 (j = 1, 2)$ , X and Y such that the following LMIs hold

[\begin{matrix} Γ_{11} & Γ_{12} & Γ_{13} & Γ_{14} & Γ_{15} & P \bar{B} & {\sqrt{τ}}_{2} {\bar{A}}^{T} P & \sqrt{ν} {\bar{A}}^{T} P & 0 & 0 \\ * & Γ_{22} & Γ_{23} & Γ_{24} & Γ_{25} & 0 & 0 & 0 & 0 & 0 \\ * & * & Γ_{33} & Γ_{34} & Γ_{35} & 0 & {\sqrt{τ}}_{2} {\bar{A}}_{1}^{T} P & \sqrt{ν} {\bar{A}}_{1}^{T} P & Γ_{39} & Γ_{3, 10} \\ * & * & * & Γ_{44} & Γ_{45} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & Γ_{55} & 0 & {\sqrt{τ}}_{2} {\bar{A}}_{2}^{T} P & \sqrt{ν} {\bar{A}}_{2}^{T} P & Γ_{59} & Γ_{5, 10} \\ * & * & * & * & * & - α I & {\sqrt{τ}}_{2} {\bar{B}}^{T} P & \sqrt{ν} {\bar{B}}^{T} P & 0 & 0 \\ * & * & * & * & * & * & - {PR}_{1}^{- 1} P & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - {PR}_{2}^{- 1} P & 0 & 0 \\ * & * & * & * & * & * & * & * & Γ_{9, 9} & 0 \\ * & * & * & * & * & * & * & * & * & Γ_{10, 10} \end{matrix}] < 0

(14)

[\begin{matrix} - α P & 0 & {\bar{C}}^{T} \\ * & - (γ - α) I & 0 \\ * & * & - γ I \end{matrix}] < 0,

(15)

α Q_{1} - (1 - α τ_{1}) Q_{2} < 0,

(16)

Q_{2} - (1 - α τ_{2}) R_{1} < 0,

(17)

[\begin{matrix} - X & - Y \\ * & - Q_{2} \end{matrix}] < 0,

(18)

where

X = [\begin{matrix} X_{1} & X_{2} & X_{3} & X_{4} & X_{5} \\ * & X_{6} & X_{7} & X_{8} & X_{9} \\ * & * & Z_{1} & Z_{2} & Z_{3} \\ * & * & * & Z_{4} & Z_{5} \\ * & * & * & * & Z_{6} \end{matrix}], Y = [\begin{matrix} Y_{1} \\ Y_{2} \\ Y_{3} \\ Y_{4} \\ Y_{5} \end{matrix}], Γ_{24} = τ_{2} X_{8}

Γ_{11} = α P + P \bar{A} + {\bar{A}}^{T} P + Q_{1} + τ_{1} Q_{2} + η Q_{3} + τ_{2} X_{1} + Y_{1} + Y_{1}^{T},

Γ_{12} = τ_{2} X_{2} + Y_{2}^{T}, Γ_{13} = P {\bar{A}}_{1} G + τ_{2} X_{3} - Y_{1} + Y_{3}^{T}, Γ_{14} = τ_{2} X_{4} + Y_{4}^{T},

Γ_{15} = P {\bar{A}}_{2} G + τ_{2} X_{5} + Y_{5}^{T}, Γ_{22} = - Q_{1} + τ_{2} X_{6}, Γ_{23} = τ_{2} X_{7} - Y_{2},

Γ_{33} = τ_{2} Z_{1} - Y_{3} - Y_{3}^{T} + σ C^{T} Φ C, Γ_{34} = τ_{2} Z_{2} - Y_{4}^{T}, Γ_{35} = τ_{2} Z_{3} - Y_{5}^{T},

Γ_{44} = τ_{2} Z_{4}, Γ_{45} = τ_{2} Z_{5}, Γ_{55} = τ_{2} Z_{6} - Φ, Γ_{25} = τ_{2} X_{9},

Γ_{39} = [\sqrt{τ_{2}} δ_{1} G^{T} {\hat{A}}_{1 l}^{T} P \dots \sqrt{τ_{2}} δ_{k} G^{T} {\hat{A}}_{1 k}^{T} P], Γ_{3, 10} = [\sqrt{ν} δ_{1} G^{T} {\hat{A}}_{1 l}^{T} P \dots \sqrt{ν} δ_{k} G^{T} {\hat{A}}_{1 k}^{T} P],

Γ_{59} = [\sqrt{τ_{2}} δ_{1} G^{T} {\hat{A}}_{2 l}^{T} P \dots \sqrt{τ_{2}} δ_{k} G^{T} {\hat{A}}_{2 k}^{T} P], Γ_{5, 10} = [\sqrt{ν} δ_{1} G^{T} {\hat{A}}_{2 l}^{T} P \dots \sqrt{ν} δ_{k} G^{T} {\hat{A}}_{2 k}^{T} P],

Γ_{9, 9} = diag \underset{k}{\underset{︸}{{- {PR}_{1}^{- 1} P, \dots, - {PR}_{1}^{- 1} P}}}, Γ_{10, 10} = diag \underset{k}{\underset{︸}{{- {PR}_{2}^{- 1} P, \dots, - {PR}_{2}^{- 1} P}}},

{\hat{A}}_{1 l} = {[\begin{matrix} 0 & {(B_{f} Ξ_{l} C)}^{T} \end{matrix}]}^{T}, {\hat{A}}_{2 l} = {[\begin{matrix} 0 & - {(B_{f} Ξ_{l})}^{T} \end{matrix}]}^{T}, l = 1, 2, \dots, k .

Proof: Choose the delay-dependent Lyapunov-Krasovskii functional in the following forms

\begin{matrix} V (ξ (t), t) = ξ^{T} (t) P ξ (t) + \int_{t - τ_{1}}^{t} ξ^{T} (s) Q_{1} ξ (s) ds \\ + \int_{t - τ_{1}}^{t} \int_{s}^{t} ξ^{T} (θ) Q_{2} ξ (θ) d θ ds \\ + η [\int_{t - ν}^{t} ξ^{T} (s) Q_{3} ξ (s) ds + \int_{t - ν}^{t} \int_{s}^{t} {\overset{\cdot}{ξ}}^{T} (θ) R_{2} \overset{\cdot}{ξ} (θ) d θ ds] \\ + \int_{t - τ_{2}}^{t} \int_{s}^{t} {\overset{\cdot}{ξ}}^{T} (θ) R_{1} \overset{\cdot}{ξ} (θ) d θ ds \end{matrix}

with $ν = τ (t) - τ_{1}, η = τ_{2} - τ (t)$ .

Along the solution of the filtering error system $\bar{Σ}$ in (13) can be obtained as

\begin{matrix} E {(ξ (t), t)} = \\ E {2 ξ^{T} (t) P \overset{\cdot}{ξ} (t)} \\ + E {ξ^{T} (t) (Q_{1} + τ_{1} Q_{2} + η Q_{3} + τ_{2} {\bar{A}}^{T} R_{1} \bar{A} + ν {\bar{A}}^{T} R_{2} \bar{A}) ξ (t)} \\ - E {ξ^{T} (t - τ_{1}) Q_{1} ξ (t - τ_{1}) + {\overset{\cdot}{ξ}}^{T} (t) (τ_{2} R_{1} + ν R_{2}) \overset{\cdot}{ξ} (t)} \\ - E {\int_{t - τ_{1}}^{t} ξ^{T} (s) Q_{2} ξ (s) ds - \int_{t - τ_{2}}^{t} {\overset{\cdot}{ξ}}^{T} (s) R_{1} \overset{\cdot}{ξ} (s) ds} \\ - E {\int_{t - ν}^{t} ξ^{T} (s) Q_{3} ξ (s) ds - \int_{t - ν}^{t} \int_{s}^{t} {\overset{\cdot}{ξ}}^{T} (θ) R_{2} \overset{\cdot}{ξ} (θ) d θ ds} \end{matrix}

(19)

Define ${\bar{Ψ}}^{T} (t) = [\begin{matrix} ξ^{T} (t) & ξ^{T} (t - τ_{1}) & ξ^{T} (t - τ (t)) & ξ^{T} (t - τ_{2}) & e_{k}^{T} (t) \end{matrix}]$ , and using free weighing matrix method combined with the Newton-Leibnitz formula, we have

2 {\bar{Ψ}}^{T} (t) N [ξ (t) - ξ (t - τ (t)) - \int_{t - τ (t)}^{t} \overset{\cdot}{ξ} (s) ds] = 0

(20)

which together with the Moon’s inequality in yields

τ_{2} {\bar{Ψ}}^{T} (t) X \bar{Ψ} (t)) + 2 {\bar{Ψ}}^{T} (t) Y [ξ (t) - ξ (t - τ (t))] > - \int_{t - τ_{2}}^{t} {\overset{\cdot}{ξ}}^{T} (s) R_{1} \overset{\cdot}{ξ} (s) ds]

(21)

According to the inequality (16), noting that

\int_{t - τ_{1}}^{t} \int_{s}^{t} ξ^{T} (θ) (Q_{2} - α Q_{1}) ξ (θ) d θ ds \leq τ_{1} \int_{t - τ_{1}}^{t} ξ^{T} (s) (Q_{2} - α Q_{1})] ξ (s) ds

(22)

For $t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$ , according to the inequality (6) and equation (10), we have

e_{k}^{T} (t) Φ e_{k} (t) \leq σ {\hat{y}}^{T} (t - τ (t)) Φ \hat{y} (t - τ (t)) = σ x^{T} (t - τ (t)) C^{T} Φ Cx (t - τ (t))

(23)

From what has been discussed above, by adding equation (20), $[e_{k}^{T} (t) Φ e_{k} (t) - e_{k}^{T} (t) Φ e_{k} (t)]$ to the right of equation (19) and combining to the inequalities (17) and (21), one has the following results

E {£ (ξ (t), t)} \leq Ψ^{T} (t) Π Ψ (t) + α ω^{T} (t) ω (t) - α E {V (ξ (t), t)}

(24)

where

Ψ^{T} (t) = [\begin{matrix} ξ^{T} (t) & ξ^{T} (t - τ_{1}) & ξ^{T} (t - τ (t)) & ξ^{T} (t - τ_{2}) & e_{k}^{T} (t) & ω^{T} (t) \end{matrix}],

Π = [\begin{matrix} Γ_{11} & Γ_{12} & Γ_{13} & Γ_{14} & Γ_{15} & P \bar{B} \\ * & Γ_{22} & Γ_{23} & Γ_{24} & Γ_{25} & 0 \\ * & * & Γ_{33} & Ξ_{34} & Γ_{35} & 0 \\ * & * & * & Γ_{44} & Γ_{45} & 0 \\ * & * & * & * & Γ_{55} & 0 \\ * & * & * & * & * & - α I \end{matrix}] + [\begin{matrix} {\bar{A}}^{T} \\ 0 \\ {\bar{A}}_{1}^{T} \\ 0 \\ {\bar{A}}_{2}^{T} \\ {\bar{B}}^{T} \end{matrix}] (τ_{2} R_{1} + ν R_{2}) {[\begin{matrix} {\bar{A}}^{T} \\ 0 \\ {\bar{A}}_{1}^{T} \\ 0 \\ {\bar{A}}_{2}^{T} \\ {\bar{B}}^{T} \end{matrix}]}^{T}

The inequality (14) guarantees $Π < 0$ , and we can further obtain from the inequality (24) that

E {(ξ (t), t)} \leq α ω^{T} (t) ω (t) - α E {V (ξ (t), t)}

(25)

For $ω (t) = 0$ and all $Ψ (t) \neq 0$ , the inequality (25) means $E {(ξ (t), t)} < 0$ . So, we can conclude that the filtering error system $\bar{Σ}$ in (13) is mean-square asymptotically stable.

Define $℧ = {ξ : E {V (ξ (t), t)} \leq 1}$ . For the state $ξ (t)$ of the filtering error system $\bar{Σ}$ in (13) with $∥ ω (t) ∥_{L_{\infty}} \leq 1$ and zero initial condition, we will discuss the inequality (25) under the following two conditions:

(i) For $E {(ξ (t), t)} \geq 0$ , it is easy to show from the inequality (25) that

E {V (ξ (t), t)} < ω^{T} (t) ω (t) \leq 1

(ii) For $E {(ξ (t), t)} < 0$ , owing to $E {V (ξ (t), t)} |_{t = 0} = 0$ , when $t > 0$ , that

E {V (ξ (t^{+}), t^{+})} < E {V (ξ (t), t)}

it is easy to show that $E {V (ξ (t), t)}$ will never reach 1 if $E {V (ξ (t), t)}$ is less than 1 under the first condition.

According to the above two cases, we conclude that ℧ is an invariant set.

Furthermore, we will establish $L_{1}$ performance for the filtering error system $\bar{Σ}$ in (13).

\begin{matrix} J = \frac{1}{γ} E {{\tilde{z}}^{T} (t) \tilde{z} (t)} - E {α ξ^{T} (t) P ξ (t) + (γ - α) ω^{T} (t) ω (t)} \\ = E {{[\begin{matrix} ξ (t) \\ ω (t) \end{matrix}]}^{T} W [\begin{matrix} ξ (t) \\ ω (t) \end{matrix}]} \end{matrix}

(26)

where

W = [\begin{matrix} \frac{1}{γ} {\bar{C}}^{T} C - α P & 0 \\ * & - (γ - α) I \end{matrix}],

the inequality (15) implies $W < 0$ , therefore

E {{\tilde{z}}^{T} (t) \tilde{z} (t)} < γ E {[α ξ^{T} (t) P ξ (t) + (γ - α) ω^{T} (t) ω (t)]}

(27)

Since ℧ is an invariant set, which means $E {ξ^{T} (t) P ξ (t)} < 1$ , for all $∥ ω (t) ∥_{L_{\infty}} \leq 1$ , it can be concluded from the inequality (27) that $E {{\tilde{z}}^{T} (t) \tilde{z} (t)} < γ^{2}$ . Furthermore, we obtain $sup_{∥ ω (t) ∥_{L_{\infty}} \leq 1} E {∥ \tilde{z} (t) ∥}_{L_{\infty}} < γ$ , and the proof is completed.

Remark 2: In many engineering fields, the systems sustain persistent heavy noise disturbance, such as: wind shear stress imposed upon aircraft wings; sea wave clutter interference signals on offshore platform, and so forth; the traditional Kalman filtering; or $H_{\infty}$ or $L_{2}$ - $L_{\infty}$ filtering can not receive good noise suppression effect. Theorem 1 proposed a valuable $L_{1}$ norm performance criterion for NCSs with sensor faults. Compared with Li et al. (2016b), the Lyapunov-Krasovskii functional with piecewise-linear sawtooth structure characteristic of transmission delay is adopted in the proof of Theorem 1, which is of general significance. It is worth emphasizing that the introduction of the DETCS makes the $L_{1}$ norm performance criterion is established more complex, and requires a certain mathematics skill, $L_{1}$ performance criterion is constructed by using flexibly free weighing matrix method, the Newton-Leibnitz formula, the integral inequality (22), and Lyapunov-Krasovskii functional and its derivative terms.

Remark 3: According to Theorem 1, the $Γ_{11} < 0$ is needed in the inequality (14) and the solution of the inequalities (14)–(18) relies on the proper choice of variable $α$ . Then, $α$ must lie inside the following interval to ensure a positive definite solution to the inequality (14), that is

0 < α < - 2 max R e (λ (\bar{A}))

Therefore, we can perform a linear search on the variable $α$ to get tighter bounds, and then obtain the minimization of the upper bound $γ$ to the $L_{1}$ gain. However, the search interval of $α$ can be reduced or $α$ given as a constant according to actual needs, thus improving the computational efficiency

Remark 4: The LMIs in Theorem 1 contain abundant system information and communication information, such as uncertainties of system parameters, sensor faults, transmission delay, and the DETCS for saving network resources. Therefore, the relation between the robust $L_{1}$ filtering and the DETCS has been established for NCS, which lays the groundwork for subsequent co-design.

Co-design between the robust $L_{1}$ filter and the DETCS

With the help of Theorem 1, now we are ready to solve the problem of $L_{1}$ filter design for uncertain NCS under DETCS.

Theorem 2: Given positive scalars $α, τ_{1}, τ_{2}, η, ν, σ, δ$ , under the DETCS, the filtering error system $\bar{Σ}$ in (13) is mean-square asymptotically stable with an $L_{1}$ performance level $γ$ if there exist matrices $U > 0$ , $V > 0$ , $\bar{Φ} > 0$ , ${\bar{Q}}_{i} > 0 (i = 1, 2, 3)$ , ${\bar{R}}_{j} > 0 (j = 1, 2)$ , ${\bar{X}}_{a} (a = 1, \dots, 9)$ , ${\bar{Y}}_{b} (b = 1, \dots, 5)$ , ${\bar{Z}}_{c} (c = 1, \dots, 6)$ , ${\bar{A}}_{f}$ , ${\bar{B}}_{f}$ , ${\bar{C}}_{f}$ and scalar $ϵ$ such that the following LMIs hold

[\begin{matrix} {\bar{Γ}}_{11} & {\bar{Γ}}_{12} & {\bar{Γ}}_{13} & {\bar{Γ}}_{14} & {\bar{Γ}}_{15} & {\bar{Γ}}_{16} & {\bar{Γ}}_{17} & {\bar{Γ}}_{18} & 0 & 0 & UH \\ * & {\bar{Γ}}_{22} & {\bar{Γ}}_{23} & {\bar{Γ}}_{24} & {\bar{Γ}}_{25} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & {\bar{Γ}}_{33} & {\bar{Γ}}_{34} & {\bar{Γ}}_{35} & 0 & {\bar{Γ}}_{37} & {\bar{Γ}}_{38} & {\bar{Γ}}_{39} & {\bar{Γ}}_{3, 10} & 0 \\ * & * & * & {\bar{Γ}}_{44} & {\bar{Γ}}_{45} & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & {\bar{Γ}}_{55} & 0 & {\bar{Γ}}_{57} & {\bar{Γ}}_{58} & {\bar{Γ}}_{59} & {\bar{Γ}}_{5, 10} & 0 \\ * & * & * & * & * & - α I & {\bar{Γ}}_{67} & {\bar{Γ}}_{68} & 0 & 0 \\ * & * & * & * & * & * & {\bar{Γ}}_{77} & 0 & 0 & 0 & UH \\ * & * & * & * & * & * & * & {\bar{Γ}}_{88} & 0 & 0 & VH \\ * & * & * & * & * & * & * & * & {\bar{Γ}}_{9, 9} & 0 \\ * & * & * & * & * & * & * & * & * & {\bar{Γ}}_{10, 10} & 0 \\ * & * & * & * & * & * & * & * & * & * & - ϵ I \end{matrix}] < 0

(28)

[\begin{matrix} - α U & - α V & 0 & L^{T} \\ * & - α V & 0 & - C_{f}^{T} \\ * & * & - (γ - α) I & 0 \\ * & * & * & - γ I \end{matrix}] < 0,

(29)

α {\bar{Q}}_{1} - (1 - α τ_{1}) {\bar{Q}}_{2} < 0,

(30)

{\bar{Q}}_{2} - (1 - α τ_{2}) {\bar{R}}_{1} < 0,

(31)

[\begin{matrix} - {\bar{X}}_{1} & - {\bar{X}}_{2} & - {\bar{X}}_{3} & - {\bar{X}}_{4} & - {\bar{X}}_{5} & - {\bar{Y}}_{1} \\ * & - {\bar{X}}_{6} & - {\bar{X}}_{7} & - {\bar{X}}_{8} & - {\bar{X}}_{9} & - {\bar{Y}}_{2} \\ * & * & - {\bar{Z}}_{1} & - {\bar{Z}}_{2} & - {\bar{Z}}_{3} & - {\bar{Y}}_{3} \\ * & * & * & - {\bar{Z}}_{4} & - {\bar{Z}}_{5} & - {\bar{Y}}_{4} \\ * & * & * & * & - {\bar{Z}}_{6} & - {\bar{Y}}_{5} \\ * & * & * & * & * & - {\bar{Q}}_{2} \end{matrix}] < 0,

(32)

Moreover, a desired $L_{1}$ filter is given in parameters as follows

[\begin{matrix} A_{f} & B_{f} \\ C_{f} & 0 \end{matrix}] = [\begin{matrix} V^{- 1} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{f} & {\bar{B}}_{f} \\ {\bar{C}}_{f} & 0 \end{matrix}] .

(33)

where

{\bar{Γ}}_{11} = {\hat{Γ}}_{11} + {\bar{Q}}_{1} + τ_{1} {\bar{Q}}_{2} + η {\bar{Q}}_{3} + τ_{2} {\bar{X}}_{1} + {\bar{Y}}_{1} + {\bar{Y}}_{1}^{T}, {\bar{Γ}}_{14} = τ_{2} {\bar{X}}_{4} + {\bar{Y}}_{4}^{T},

{\hat{Γ}}_{11} = [\begin{matrix} α U + UA + A^{T} U + ϵ E^{T} E & α V + {\bar{A}}_{f} + A^{T} V \\ * & α V + {\bar{A}}_{f} + {\bar{A}}_{f}^{T} \end{matrix}], {\bar{Γ}}_{17} = \sqrt{τ_{2}} [\begin{matrix} A^{T} U & A^{T} V \\ {\bar{A}}_{f}^{T} & {\bar{A}}_{f}^{T} \end{matrix}],

{\bar{Γ}}_{12} = τ_{2} {\bar{X}}_{2} + {\bar{Y}}_{2}^{T}, {\bar{Γ}}_{13} = {\hat{Γ}}_{13} + τ_{2} {\bar{X}}_{3} - {\bar{Y}}_{1} + {\bar{Y}}_{3}^{T}, {\bar{Γ}}_{15} = {\hat{Γ}}_{15} + τ_{2} {\bar{X}}_{5} + {\bar{Y}}_{5}^{T},

{\hat{Γ}}_{13} = [\begin{matrix} {\bar{B}}_{f} \bar{Ξ} C & 0 \\ {\bar{B}}_{f} \bar{Ξ} C & 0 \end{matrix}], {\hat{Γ}}_{15} = [\begin{matrix} - {\bar{B}}_{f} \bar{Ξ} & 0 \\ - {\bar{B}}_{f} \bar{Ξ} & 0 \end{matrix}], {\bar{Γ}}_{16} = [\begin{matrix} UB \\ VB \end{matrix}], {\bar{Γ}}_{18} = \sqrt{\frac{τ_{2}}{ν}} {\bar{Γ}}_{17},

{\bar{Γ}}_{22} = - {\bar{Q}}_{1} + τ_{2} {\bar{X}}_{6}, {\bar{Γ}}_{23} = τ_{2} {\bar{X}}_{7} - {\bar{Y}}_{2}, {\bar{Γ}}_{24} = τ_{2} {\bar{X}}_{8}, {\bar{Γ}}_{25} = τ_{2} {\bar{X}}_{9},

{\bar{Γ}}_{33} = τ_{2} {\bar{Z}}_{1} - {\bar{Y}}_{3} - {\bar{Y}}_{3}^{T} + σ C^{T} \bar{Φ} C, {\bar{Γ}}_{34} = τ_{2} {\bar{Z}}_{2} - {\bar{Y}}_{4}^{T}, {\bar{Γ}}_{35} = τ_{2} {\bar{Z}}_{3} - {\bar{Y}}_{5}^{T},

{\bar{Γ}}_{37} = \sqrt{τ_{2}} {\hat{Γ}}_{13}^{T}, {\bar{Γ}}_{38} = \sqrt{ν} {\hat{Γ}}_{13}^{T}, {\bar{Γ}}_{44} = τ_{2} {\bar{Z}}_{4}, {\bar{Γ}}_{45} = τ_{2} {\bar{Z}}_{5}, {\bar{Γ}}_{55} = τ_{2} {\bar{Z}}_{6} - \bar{Φ},

{\bar{Γ}}_{39} = [\sqrt{τ_{2}} δ_{1} {\tilde{A}}_{11} \dots \sqrt{τ_{2}} δ_{k} G^{T} {\tilde{A}}_{1 k}], {\bar{Γ}}_{3, 10} = [\sqrt{ν} δ_{1} {\tilde{A}}_{11} \dots \sqrt{ν} δ_{k} {\tilde{A}}_{1 k}],

{\bar{Γ}}_{57} = \sqrt{τ_{2}} {\hat{Γ}}_{15}^{T}, {\bar{Γ}}_{58} = \sqrt{ν} {\hat{Γ}}_{15}^{T}, {\bar{Γ}}_{67} = \sqrt{τ_{2}} {\bar{Γ}}_{16}^{T}, {\bar{Γ}}_{68} = \sqrt{ν} {\bar{Γ}}_{16}^{T},

{\bar{Γ}}_{59} = [\sqrt{τ_{2}} δ_{1} {\tilde{A}}_{21} \dots \sqrt{τ_{2}} δ_{k} {\tilde{A}}_{2 k}], Γ_{5, 10} = [\sqrt{ν} δ_{1} {\tilde{A}}_{21} \dots \sqrt{ν} δ_{k} {\tilde{A}}_{2 k}],

{\bar{Γ}}_{77} = - 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{1}, {\bar{Γ}}_{88} = - 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{2},

{\bar{Γ}}_{9, 9} = diag \underset{k}{\underset{︸}{{- 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{1}, \dots, - 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{1}}}},

{\bar{Γ}}_{10, 10} = diag \underset{k}{\underset{︸}{{- 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{2}, \dots, - 2 ε_{1} \bar{P} + ε_{1}^{2} {\bar{R}}_{2}}}},

{\tilde{A}}_{1 l} = [\begin{matrix} C^{T} Ξ_{l}^{T} {\bar{B}}_{f}^{T} & C^{T} Ξ_{l}^{T} {\bar{B}}_{f}^{T} \\ 0 & 0 \end{matrix}], {\tilde{A}}_{2 l} = [\begin{matrix} - Ξ_{l}^{T} {\bar{B}}_{f}^{T} & - Ξ_{l}^{T} {\bar{B}}_{f}^{T} \\ 0 & 0 \end{matrix}], (l = 1, 2, \dots, k) .

Proof: Owing to $(R_{i} - ε_{i}^{- 1} P) R_{1}^{- i} (R_{i} - ε_{i}^{- 1} P) \geq 0 (i = 1, 2)$ , one has $- {PR}_{i}^{- 1} P < - 2 ε_{i} P + ε_{i}^{2} R_{i}$ , substitute $- {PR}_{i}^{- 1} P$ with $- 2 ε_{i} P + ε_{i}^{2} R_{i}$ into the inequality (14).

By Theorem 1, if the inequalities (14)–(18) hold, the matrix P is nonsingular, since $P > 0$ , partition P as

P = [\begin{matrix} P_{1} & P_{2} \\ P_{2}^{T} & P_{3} \end{matrix}]

(34)

As we are considering a full-order filter, $P_{2}$ is a square. Without loss of generality, we suppose $P_{2}$ is nonsingular. Define the following matrices which are also nonsingular

W = [\begin{matrix} I & 0 \\ 0 & P_{3}^{- 1} P_{2}^{T} \end{matrix}], U = P_{1}, V = P_{2} P_{3}^{- 1} P_{2}^{T}

(35)

and

[\begin{matrix} {\bar{A}}_{f} & {\bar{B}}_{f} \\ {\bar{C}}_{f} & 0 \end{matrix}] = [\begin{matrix} P_{2} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} A_{f} & B_{f} \\ C_{f} & 0 \end{matrix}] [\begin{matrix} P_{3}^{- 1} P_{2}^{T} & 0 \\ 0 & I \end{matrix}]

(36)

Let $J_{1} = diag {W, W, W, W, W, I, W, W, \underset{2 k}{\underset{︸}{W, \dots, W}}}, J_{2} = diag {W, I, I}, J_{3} = W, J_{4} = W, J_{5} = diag {W, W}$ , performing congruence transformations to the inequalities (14)–(18) by matrices $J_{1}, J_{2}, J_{3}, J_{4}$ and $J_{5}$ , respectively. Defining ${\bar{Q}}_{i} = W Q_{i} W^{T} (i = 1, 2, 3)$ , ${\bar{R}}_{j} = W R_{j} W^{T} (j = 1, 2)$ , ${\bar{X}}_{a} = W X_{a} W^{T} (a = 1, \dots, 9)$ , ${\bar{Y}}_{b} = W Y_{b} W^{T} (b = 1, \dots, 5)$ , ${\bar{Z}}_{c} = W Z_{c} W^{T} (c = 1, \dots, 6)$ , $\bar{ϕ} = W ϕ W^{T}$ , and using Lemma 1, we can obtain readily the inequalities (28)–(32).

Equation (36) is equivalent to

\begin{matrix} [\begin{matrix} A_{f} & B_{f} \\ C_{f} & 0 \end{matrix}] = [\begin{matrix} P_{2}^{- 1} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{f} & {\bar{B}}_{f} \\ {\bar{C}}_{f} & 0 \end{matrix}] [\begin{matrix} P_{2}^{- T} P_{3} & 0 \\ 0 & I \end{matrix}] \\ = [\begin{matrix} {(P_{2}^{- T} P_{3})}^{- 1} V^{- 1} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{f} & {\bar{B}}_{f} \\ {\bar{C}}_{f} & 0 \end{matrix}] [\begin{matrix} P_{2}^{- T} P_{3} & 0 \\ 0 & I \end{matrix}] \end{matrix}

(37)

And note that the filter matrices $A_{f}, B_{f}$ and $C_{f}$ in equation (4) can be written as the equality (37), which implies that $P_{2}^{- T} P_{3}$ can be viewed as similarity transformation on the state-space realization of the filter, and as such, has no effect on the filter mapping from $\hat{y} (t)$ to $z_{f} (t)$ . Without loss of generality, we may set $P_{2}^{- T} P_{3} = I$ , and thus obtain equation (33). The proof is completed.

Remark 5: Once the values of transmission delay-related parameters $τ_{1}, τ_{2}, η, ν, h, σ, δ$ shown in Theorem 2 are chosen properly, the existence conditions for the event-triggered filters become strict LMIs, which are convex in the scalar $γ$ . Selecting the scalar $γ$ as an optimization variable, and the minimized $L_{1}$ performance index $γ$ can be solved in the following convex optimization problem

\min γ subject to the inequalities (28) - (32)

(38)

and the admissible event-triggered $L_{1}$ filter parameters can be obtained by equation (33).

Numerical example and result analysis

Numerical example

Suppose there exist two sensors (sensor 1 and sensor 2), that is $k = 2$ , the parameter matrices of the system $Σ$ in (1) are as follows

A = [\begin{matrix} - 2.8 & 0.2 \\ 0.3 & - 0.3 \end{matrix}], B = [\begin{matrix} 0.1 \\ - 0.08 \end{matrix}], C = [\begin{matrix} - 1.1 & 0.5 \\ 0.5 & - 1.2 \end{matrix}], L = [\begin{matrix} 1 & - 0.5 \end{matrix}],

E = [\begin{matrix} 0.1 & 0 \\ 0.1 & 0.1 \end{matrix}], H = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}] .

Assuming that the networked-related parameters as follows: $h = 0.1, τ_{1} = 0.01, τ_{2} = 0.05, η = 0.02, ν = 0.02$ , the initial states are chosen as $x (0) = [1 - 1]^{T}, x_{f} (0) = [0 0]^{T}$ , the persistent and amplitude-bounded disturbance input $ω (t) = \sin (0.08 π t)$ . Then, for two working modes of sensors: (i) the sensors work normally, that is, $Ξ = I$ , (ii) the sensors fail partially, the fault modes follow the normal distribution of ${\bar{Ξ}}_{1} = 0.25, δ_{1} = 0.05$ and ${\bar{Ξ}}_{2} = 0.75, δ_{2} = 0.05$ , respectively. (The sensor fault mode curves are shown in Figure 2). We will discuss the robust $L_{1}$ filter design for the NCS $Σ$ in (1) under the following three cases.

Figure 2.

Sensor fault mode curves.

Case 1: The NCS $Σ$ in (1) is time-triggered, setting $σ = 0$ in inequality (6).

(i) The sensors work normally: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.4498$ , the corresponding filter parameters

A_{f} = [\begin{matrix} - 2.5977 & 2.1324 \\ 1.9607 & - 1.8528 \end{matrix}], B_{f} = [\begin{matrix} - 35.5465 & - 52.1781 \\ 12.7611 & 27.8565 \end{matrix}],

C_{f} = [\begin{matrix} - 0.0185 & 0.0156 \end{matrix}] .

(ii) The sensors fail partially: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.7008$ , the corresponding filter parameters

A_{f} = [\begin{matrix} - 2.5930 & 2.1570 \\ 1.9871 & - 1.8346 \end{matrix}], B_{f} = [\begin{matrix} - 73.0118 & - 109.5180 \\ 36.6844 & 67.9857 \end{matrix}],

C_{f} = [\begin{matrix} - 0.0170 & 0.0142 \end{matrix}] .

and the filter estimation and estimation error responses of the system generated with PTTCS are shown in Figure 3, from which we can see that though the initial condition of the system is nonzero, the estimation error responses of the system will eventually fluctuate at zero when the sensors work normally or fail partially, therefore, the designed $L_{1}$ filter can measure effectively the signal to be estimated of the system for all persistent and amplitude-bounded disturbances. However, the results become less preferable as we use PTTCS, which may lead to unnecessary data transmissions and waste communication resources.

Figure 3.

Filter estimation and estimation error responses of the system for Case 1.

Case 2: The NCS $Σ$ in (1) is event-triggered, and the corresponding trigger parameter $σ = 0.3$ in the inequality (6).

(i) The sensors work normally: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.4485$ , the corresponding trigger matrix and filter parameters

Φ = [\begin{matrix} 1.0506 & 1.3603 \\ 1.3603 & 2.3119 \end{matrix}], A_{f} = [\begin{matrix} - 2.6074 & 2.1401 \\ 1.9684 & - 1.8590 \end{matrix}],

B_{f} = [\begin{matrix} - 29.6627 & - 45.6988 \\ 8.1597 & 23.7004 \end{matrix}], C_{f} = [\begin{matrix} - 0.0182 & 0.0155 \end{matrix}] .

Figure 4 displays the transmission instants and release intervals of system, it is clear that only the sampled-data satisfying condition in the inequality (6) is transmitted, so the bandwidth occupation of network communication can be reduced effectively.

Figure 4.

Transmission instants and release intervals $σ = 0.3$ .

(ii) The sensors fail partially: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.6990$ , the corresponding trigger matrix and filter parameters

Φ = [\begin{matrix} 2.7224 & - 1.6201 \\ - 1.6201 & 4.9872 \end{matrix}], A_{f} = [\begin{matrix} - 2.6005 & 2.1601 \\ 1.9931 & - 1.8372 \end{matrix}],

B_{f} = [\begin{matrix} - 61.3720 & - 94.7952 \\ 27.2320 & 57.7568 \end{matrix}], C_{f} = [\begin{matrix} - 0.0172 & 0.0145 \end{matrix}] .

Also, the filter estimation and estimation error responses of the system generated with DETCS are shown in Figure 5. From Figure 4 and Figure 5, we can see that the co-design scheme proposed in this paper can not only save the communication resources effectively, but also measure the signal to be estimated of the system significantly.

Figure 5.

Filter estimation and estimation error responses of the system for Case 2.

Case 3: More and more unmanned surface vehicles (UAVs) have been developed for military, mobile robots and environmental monitoring research due to their advantages of low cost, small size and powerful field operational. However, the USV systems are prone to wave erosion, seawater erosion, which will inevitably lead to loss effectiveness of actuator, saturation etc and affect system performance. To further show the effectiveness of the obtained $L_{1}$ filter under DETCS and the capability of disturbances suppression more effectively, considering an unmanned surface vehicle (USV), the corresponding motion equations are described as follows (Wang et al., 2019b)

{\begin{matrix} \overset{\cdot}{υ} (t) = - \frac{1}{T_{υ}} υ (t) + \frac{K_{d υ}}{T_{υ}} θ (t), \\ \overset{\cdot}{r} (t) = \frac{K_{υ r}}{T_{r}} υ (t) - \frac{1}{T_{r}} r (t) + \frac{K_{dr}}{T_{r}} θ (t) + \frac{1}{T_{r}} ω_{ψ} (t), \\ \overset{\cdot}{p} (t) = ω_{n}^{2} K_{υ p} υ (t) - 2 ζ ω_{n} p (t) - ω^{2} φ (t) + ω_{n}^{2} K_{dp} θ (t) + ω_{n}^{2} ω_{φ} (t), \\ \overset{\cdot}{φ} (t) = p (t), \\ \overset{\cdot}{ψ} (t) = r (t) . \end{matrix}

(39)

where $υ (t), r (t), ψ (t), p (t), φ (t)$ and $θ (t)$ denote the sway velocity caused by the rudder motion, the yaw velocity, the heading angle, the roll velocity, the roll angle, the rudder angle, respectively; $ω_{ψ} (t)$ and $ω_{φ} (t)$ denote the wave-induced disturbance on $ψ (t)$ and $φ (t)$ ; $K_{υ r}, K_{υ p}, K_{d υ}, K_{dr}$ and $K_{dp}$ denote the given gains; $T_{υ}$ and $T_{r}$ denote the time constants; $ζ$ and $ω_{n}$ denote the damping coefficient and the undamped natural frequency, respectively. Define the system state vector $x (t) = col {υ (t), r (t), ψ (t), p (t), φ (t)}$ with $x (t) \in ℜ^{n}$ , the disturbance vector as $ω (t) = col {ω_{ψ} (t), ω_{φ} (t)}$ with $ω (t) \in ℜ^{d}$ , the system input as $u (t) = θ (t)$ with $u (t) \in ℜ^{m}$ . Then, the dynamic equation (39) can be rewritten as $Σ$ in (1), where the practical parameters of an USV are listed (Wang et al., 2019b)

T_{υ} = 0.5263, T_{r} = 0.4211, K_{dr} = - 0.0103, K_{dp} = - 0.0202, K_{d υ} = 0.0380,

K_{υ p} = 0.7980, K_{υ r} = - 0.4600, ω_{n} = 1.6300, ζ = 2.0840 .

then, one has

A = [\begin{matrix} - 1.9 & 0 & 0 & 0 & 0 \\ - 1.0925 & - 2.3750 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 2.1202 & 0 & 0 & - 6.7938 & - 2.6569 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}], B = [\begin{matrix} 0.0722 \\ - 0.0244 \\ 0 \\ - 0.0537 \\ 0 \end{matrix}],

C = [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 2.3750 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2.6569 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}], L = [\begin{matrix} 1 & 0.8 & 1 & - 1 & 0.6 \end{matrix}] .

(i) The sensors work normally: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.3570$ , the corresponding trigger matrix and filter parameters

ϕ = [\begin{matrix} 1.1231 & 3.1571 & - 0.0016 & 0.3031 & 0.2546 \\ 3.1571 & 2.8642 & 0.1616 & - 2.2433 & 0.1357 \\ - 0.0016 & 0.1616 & 4.0694 & - 0.0870 & 0.0147 \\ 0.3031 & - 2.2433 & - 0.0870 & 1.0195 & 0.2739 \\ 0.2546 & 0.1357 & 0.0147 & 0.2739 & 1.9380 \end{matrix}],

A_{f} = [\begin{matrix} - 1.8609 & 0.6669 & - 0.0068 & 1.4132 & - 0.0037 \\ 0.5791 & - 0.3734 & 0.0024 & - 0.4773 & 0.0020 \\ 0.0002 & 0.0001 & - 0.1470 & 0.0000 & - 0.0001 \\ 1.2747 & - 0.4958 & 0.0052 & - 1.1985 & 0.0019 \\ - 0.0001 & 0.0008 & - 0.0001 & - 0.0007 & - 0.1474 \end{matrix}],

B_{f} = [\begin{matrix} 0.0018 & - 1.1053 & - 0.0003 & - 11.2436 & 0.0009 \\ 0.0056 & 8.4561 & - 0.0001 & 4.0167 & 0.0049 \\ - 0.0025 & - 1.2411 & 0.0000 & - 0.2853 & - 0.0016 \\ - 0.0075 & 18.4895 & 0.0003 & 16.8820 & 0.0034 \\ 0.0014 & - 0.9504 & 0.0000 & 0.1814 & 0.0002 \end{matrix}],

C_{f} = [\begin{matrix} - 0.0191 & 0.0065 & - 0.0003 & 0.0166 & - 0.0004 \end{matrix}] .

(ii) The sensors fail partially: according to Theorem 2, the optimal $L_{1}$ performance criterion is found to be $γ^{*} = 0.7239$ , the corresponding filter parameters

ϕ = [\begin{matrix} 1.3490 & 4.0419 & - 0.0072 & 1.3135 & 0.2169 \\ 4.0419 & 3.2120 & 0.1482 & - 2.8767 & 0.1029 \\ - 0.0072 & 0.1482 & 2.6719 & - 0.0829 & 0.0168 \\ 1.3135 & - 2.8767 & - 0.0829 & 3.3695 & 0.2411 \\ 0.2169 & 0.1029 & 0.0168 & 0.2411 & 2.5687 \end{matrix}],

A_{f} = [\begin{matrix} - 1.8525 & 0.6840 & - 0.0062 & 1.4461 & - 0.0037 \\ 0.5953 & - 0.3223 & 0.0022 & - 0.4886 & 0.0015 \\ 0.0001 & 0.0000 & - 0.0908 & - 0.0000 & - 0.0000 \\ 1.3104 & - 0.5087 & 0.0047 & - 1.1664 & 0.0024 \\ - 0.0000 & 0.0003 & - 0.0000 & - 0.0003 & - 0.0909 \end{matrix}],

B_{f} = [\begin{matrix} 0.0423 & - 50.0247 & - 0.0005 & - 79.7773 & - 0.0065 \\ 0.0091 & 38.7283 & 0.0002 & 26.8031 & 0.0151 \\ - 0.0079 & - 3.3002 & - 0.0001 & - 0.6640 & - 0.0041 \\ - 0.0645 & 85.1468 & - 0.0001 & 82.2457 & 0.0134 \\ 0.0051 & - 3.3824 & 0.0001 & 0.4323 & 0.0002 \end{matrix}],

C_{f} = [\begin{matrix} - 0.0166 & 0.0050 & - 0.0003 & 0.0150 & - 0.0005 \end{matrix}] .

and the filter estimation and estimation error responses of the system generated with DETCS are shown in Figure 6, from which we can see that the proposed $L_{1}$ filter design method has effectiveness and usefulness to solve the problem of actual systems, and offers a valuable method for solving systems with multiple sensor fault modes and persistent heavy noise disturbance, such as, the petrochemical process with inlet flow fluctuation, the longitudinal control system of four-rotor aircraft, and so forth.

Figure 6.

Filter estimation and estimation error responses of the system for Case 3.

Result analysis of DETCS

In order to better illustrate the results of this paper, taking the partial failure of sensors in Case 2 as an example.

When the optimal interference suppression level $γ = 0.6990$ , data transmission under different event-triggered thresholds $σ$ is shown in Table 1, where the n is the number of the transmitted sampled-data; the r is the ratio of transmitted data under DETCS to transmitted data under PTTCS; the $\bar{h}$ represents average transmission periods. It is seen from Table 1 that only the sampled-data satisfying the condition in the inequality (6) is transmitted, a stricter trigger condition will result in a longer average transmission period, and a less amount of send data as well, which is more obvious effect of saving network resources. Nevertheless, increasing the value of event-triggered parameter blindly can cause the system performance degradation or even instability. Thus, by adjusting trigger parameter $σ$ , the coordination between network resource occupation and system performance can be achieved.

Table 1.

Data transmission under different $σ$ .

$σ$	n	$r / %$	$\bar{h}$
0	500	100	0.1
0.1	457	91.4	0.1094
0.3	351	70.2	0.1425
0.5	246	49.2	0.2033

Given event-triggered threshold $σ = 0.3$ , data transmission under different disturbance suppression levels $γ$ is shown in Table 2. It is clear that the average transmission period under DETCS is much larger than that under PTTCS ( $h = 0.1 s$ ), regardless of the conditions without disturbance, with disturbance or with optimal disturbance suppression level of the system. At the same time, if the system is expected to have an optimal interference suppression level $γ = 0.6990$ , it will take up a lot of network resources, and lead to the decreasing of the network performance. Therefore, practical requirements based on system design, we can find a compromise scheme between the level of disturbance suppression of the system and the amount of network resources occupied.

Table 2.

Data transmission under different $γ$ .

$γ$	n	$r / %$	$\bar{h}$
0	238	46.5	0.2101
0.2	321	54.0	0.1558
0.8	386	61.0	0.1295
0.6990	425	72.5	0.1176

Conclusion

In this paper, the robust $L_{1}$ filtering problem has been addressed for a class of NCSs with transmission delay and multiple sensor fault modes. To reduce the network burden and the energy consumption, the DETCS is considered, which determines whether the current sampled measurement by sensors should be transmitted or not. By employing a novel Lyapunov-Krasovskii functional with sawtooth structure characteristic of an artificial delay and by using LMIs technique, a sufficient condition has been established to ensure mean-square asymptotically stable with the desired robust $L_{1}$ performance constraint for the filtering error system, and the corresponding co-design scheme has been given to obtain the parameters of the event generator and the filter. Simulation results show that the proposed method has used less communication bandwidth while preserving the desired filtering performance. The PTTCS wastes communication resources, and the DETCS may increase output error. To reduce the design conservatism and make it more applicable in practice, a better research direction will consider a hybrid time/event triggered communication scheme in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Heilongjiang Province of China (LH2020F004).

ORCID iD

Ji Qi

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Event-triggered L 1 filtering for uncertain networked control systems with multiple sensor fault modes

Abstract

Keywords

Introduction

Problem formulation

Main results

L 1 performance analysis under DETCS

Co-design between the robust L 1 filter and the DETCS

Numerical example and result analysis

Numerical example

Result analysis of DETCS

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

$L_{1}$ performance analysis under DETCS

Co-design between the robust $L_{1}$ filter and the DETCS