A novel adaptive event-triggered reliable $H_\infty$ control approach for networked control systems with actuator faults

1.   Introduction

Over the past few decades, the control issues of NCSs have attracted the attention of many researchers, where physical processes, actuators, and sensors are linked together over communication networks ^[1]. Significant research has been conducted on its wide application in a variety of potential fields, such as unmanned vehicles ^[2,3,4], load frequency control (LFC) system ^[5,6,7,8], internet of things and smart power grid.

Communication resources have proved to be increasingly important for control implementations as the medium of data transmission. As a result, many studies are bending their efforts to save communication bandwidth and reduce communication load ^[9,10,11,12]. The event-triggering control mechanism (ETM) has received more and more attention in the control engineering fields ^{[13,14,15,16,17,18,19]}. The ETM performs its control behavior as needed and is superior to stochastic and periodic sampling mechanisms, which can cause less occupied communication bandwidth. In recent years, some significant advanced achievements with regard to ETM have been discussed in many research findings, such as ^{[5,6,7,13,14,20,21,22,23,24]}. The security control problem of NCSs is addressed in ^[25], which proposes a novel resilient event-triggering strategy considering the uncertainty of triggering conditions caused by DoS attacks. An event-triggering scheme for NCSs is put forward in ^[26], where the sampling data will not be transmitted through the communication networks unless a predesigned threshold condition is violated. An ETM for multi-area power systems is proposed in ^[27]. It is worth noting that the triggering parameter in the general ETM is a constant, independent of the system fluctuation state. To conserve as much communication bandwidth as possible, the AETM with adaptive triggering conditions is widely studied ^{[6,7,28,29,30]}. A dynamically triggering algorithm for multi-areas power systems is designed by ^[6]. The triggering parameter is no longer constant but adjusted with the change of system state, which can effectively reduce the number of data packets. The main purpose of ^[7] is to develop an adaptive event-triggering control algorithm for the multi-area LFC system over a limited bandwidth communication channel. Dong et al. ^[28] puts forward a new event-triggering control architecture for LFC systems with adaptive dynamic programming (ADP). A novel dynamic AETM is proposed in ^[29], which is flexible enough because the threshold condition is variable for the system state. With the presence of hybrid attacks, a new dynamic-memory event-triggered $H_\infty$ LFC method is established in ^[30]. The ETM is an effective method to occupy less communication bandwidth and reduce the computational burden of the control system. Appropriate triggering parameters can decide the performance of an ETM design. It motivates us to design an ETM with an adaptive triggering threshold associated with the system state.

NCSs ^[31,32,33] are also widely used in a variety of applications ranging from industrial control to critical infrastructure such as power systems, transportation systems, and water supply systems. In fact, the combination of cyber and physical systems will bring some threats to the security of NCSs, such as actuator faults, packet dropout/disorder brought from cyber attacks, physical devices or human actions. Hence, it is very important to design a reliable controller for improving the security of NCSs that can resist and revive failures. There have been many published achievements with regard to reliable controller design. Wang et al.^[32] discuss the reliable $H_\infty$ and passive control for NCSs under an adaptive event-triggered scheme with actuator faults and randomly occurring nonlinear perturbations. The robust reliable control problem for uncertain networked control systems is fully investigated in ^[33]. Li et al.^[34] study the problem of stochastic reliable control for NCSs subjected to actuator failure and input saturation. From the above-published works, it is necessary to design a reliable control approach considering the failures caused by the communication network. Thus, how to design a reliable controller against actuator failures considering AETM is another motivation for this work.

There are some kinds of controller techniques to face the control system, such as predictor-observer based control techniques ^[35], multi-rate control solutions ^[36], gain scheduling control methods ^[37] and $H_\infty$ methods. To accurately estimate and account for the "total disturbance" in uncertain systems with sensor delay, the extended state predictor observer (ESPO)-based active disturbance rejection control (ADRC) is first developed in ^[35]. A periodic event-triggered system that is susceptible to external disturbance and sensor noise is discussed in ^[36] along with a systematic non-uniform multi-rate estimation and control framework. The gain-scheduling control strategy is discussed in ^[37] for an experimental setup featuring a flexible beam, whose stiffness varies with its length. Among the above control theories and methods, $H_\infty$ theory and methods are the most widely used and have received a lot of attention in engineering ^[6,7,24,32]. When the mathematical model differs from the actual controlled object, the $H_\infty$ controller can keep system performance within acceptable limits. An event-triggered load frequency control for a multi-area power system is investigated in ^[6] by setting the $H_\infty$ control performance index. The stability and stabilization criteria are studied in ^[7] to balance the required communication resources and the desired $H_\infty$ control performance. The event-triggered parameters and the memory $H_\infty$ LFC controller gains are obtained simultaneously by a sufficient stabilization criterion in ^[24]. In ^[32], the mixed $H_\infty$ and passive control for NCSs under AETM with actuator faults and nonlinear external disturbances have been investigated for the first time. These efforts inspire us to conduct research on the reliable $H_\infty$ control approach, considering actuator faults and AETM for NCSs.

The goal of this paper is to develop a novel event-triggered reliable $H_\infty$ control approach for NCSs using AETM while accounting for actuator faults. The main contributions of this paper are listed as follows:

1) Firstly, a novel AETM has been developed that dynamically adjusts the triggering threshold according to the fluctuating trend of the current sampling state. However, the triggering parameter of ETM is a constant in ^[23,26,27]. Our proposed AETM is a more general case compared with other published ETMs like ^[23,26,27].

2) Moreover, a set of independent stochastic variables varying in $[0, \varpi]$ are modeled to describe the randomly occurring actuator faults. Hence, the sampling-data model of NCSs with AETM and actuator faults is built considering packet dropout and packet disorder caused by the communication network.

3) Furthermore, the less conservative conditions for the asymptotically stable of NCSs and event-triggered reliable $H_\infty$ controller approach are obtained by adopting a novel integral inequality. It is composed by a sequence of integrals only related to the system state, which removes the involved auxiliary function. Finally, the same simulation results can verify the progressiveness of our proposed control method.

4) Finally, through case simulation, the superiority of the proposed AETM in reducing the consumption of network resources is proved under the premise of ensuring the stability of the system. At the same time, it is proved that the proposed event-triggered reliable $H_\infty$ control approach can still guarantee the $H_\infty$ stability of the system under certain actuator faults. The method proposed in this paper can also be applied to other industrial control network systems to achieve less network bandwidth occupancy and anti-interference ability against actuator faults.

The remainder of this work is organized as follows. The adaptive event-triggering mechanism, the sampling-data model of NCSs considering AETM and actuator faults, and $H_\infty$ control objectives are described in Section 2. The main results are shown in Section 3, which are $H_\infty$ stability conditions and the event-triggering $H_\infty$ reliable controller design under AETM. In Section 4, a simulation example is applied to prove the progressiveness of the proposed reliable control method under the AETM for NCSs. The last Section 5 concludes this work.

Notation: Let ${\mathbb{ R}}$, ${\mathbb{ R}^{n}}$, $\mathbb{ R} ^{n*m}$ denotes the set of real numbers, the n-dimensional Euclidean space, the set of all $n \times m$ real matrices, respectively; $P^{-1}$ and $P^T$ denote the inverse and transpose of a given matrix $P$, respectively; $P > 0$ stands for that the matrix $P$ is a real symmetric positive definite matrix; $diag\{\dots\}$ represents a block-diagonal matrix; $\mathbb{E}[.]$ represents the expectation of random variable; $*$ in the matrix represents the symmetry of matrix; $\mathcal{L}_2[0, \infty)$ denotes the space of all square-suable vectors function over $[0, \infty)$.

2.   Preliminaries

In this section, the event triggering control framework considering actuator faults and the proposed AETM under our control objectives is presented.

2.1.   Continuous-time model of NCSs

A state-space expression of NCS is proposed in Figure 1 considering exogenous disturbance can be listed as follows:

Where $x(t)\in \mathbb{R}^n$, $u^w(t)\in \mathbb{R}^m, \Delta w(t)\in \mathcal{L}_2[0, \infty)$, $y(t)\in \mathbb{R}^l$ denotes state, controlled input, disturbance input and control output vector respectively; Constant matrices with compatible dimensions are denoted by the symbols $A$, $A_d$, $B$, $C$, and $D$. $x(t_0) = x_0$ represents the initial condition for system (2.1). The NCS is controlled over a communication network by a network state feedback controller, which is shown in Figure 1. In order to simplify the analysis, the delays from ETM to Controller and from Controller to Actuator are considered as the total communication delay $\tau$.

2.2.   Adaptive event-triggering mechanism

Firstly, the traditional ETM, which is also introduced in ^[23,26,27], is listed as follows:

Where $t_kh, k\in \mathbb{R}$ denotes the latest transmitted instant determined by ETM, $l_mh = t_kh+mh, m\in \mathbb{R}$ means the sampling instant by sensor. $\Phi$ is a positive-definite matrix, $e(l_mh) = x(l_mh)-x(t_kh)$, and $\kappa$ is a preselected constant.

Compared with the time-triggered mechanism, ETM (2.2) can effectively reduce the occupation of communication bandwidth, owing to the occurrence of a transmission depends on a state-dependent threshold. In other words, the sampling data will only be sent if a certain threshold is broken. Nevertheless, it can be seen from those corresponding simulation results that there are still many transmitting packets over the communication networks even if the system is stable. We hope that when the system is well controlled, the number of transmitted packets will be reduced. From this point of view, the preselected parameter $\kappa$ should be set as an adaptive value adjusted based on the latest transmitted data and current sampled data. Based on the above supposition, the advanced AETM is proposed as follows:

Where dynamic parameter $\kappa(l_mh)$ is determined by

Where dynamic parameter $\kappa(l_{m}h)$ changes with every sampling instant. $\kappa(l_{(m-1)}h) = \kappa(t_kh+(m-1)h)$ means the last dynamic parameter in the last sampling instant $l_{(m-1)}h$. $\lambda = 1-\frac{2\rho}{\pi}atan (\xi(l_mh)), \xi(l_mh) = \frac{\mid\mid x(l_mh)\mid\mid-\mid\mid x(t_kh)\mid\mid}{\mid\mid x(t_kh)\mid\mid}$. $\rho$ is a positive integer. $atan(.)$ is the arctangent function. It's obvious that $[\kappa_{min}, \kappa_{max}]$ is the bound of $\kappa(l_{m}h)$, where $\kappa_{min}, \kappa_{max}$ are both positive integer. $\kappa(0) = \kappa_{min}$ is the initial value of $\kappa(l_mh)$. The dynamic change logic of $\kappa(l_{m}h)$ is shown in Figure 2.

Remark 1: The arctangent function atan(.) is monotonically increasing over its domain $(-\infty, \infty)$, and the range of atan(.) is $(-\frac{\pi}{2}, \frac{\pi}{2})$. $\kappa(l_{m}h)$ is dynamically adjusted in each sampling instant based on the fluctuating trend of the current sampling state. As shown in Figure 2, if the absolute value of the current sampling state is greater than the absolute value of the recently transmitted state ($\mid\mid x(l_mh)\mid\mid-\mid\mid x(t_kh)\mid\mid > 0$), it indicates that the current state is far from the equilibrium point $(x(t) = 0)$. Then $\xi(l_mh) > 0$ and $\lambda\in (1-\rho, 1)$ with $\rho$ being a positive integer. According to Eq (2.4), if $\rho\leq 1$, then $\lambda\kappa(l_{m-1}h)) < \kappa(l_{m-1}h)$ and $\lambda\kappa(l_{m-1}h)) > 0$, so $\kappa(l_{m}h)) = min (\kappa_{max}, \lambda\kappa(l_{m-1}h)) = \lambda\kappa(l_{m-1}h) < \kappa(l_{m-1}h)$, which is presented as the situation of Ⅰ in Figure 2. If $\rho > 1$, then $\lambda\kappa(l_{m-1}h))$ maybe negative. If $\lambda\kappa(l_{m-1}h))$ is positive, $\kappa(l_{m}h)) = min (\kappa_{max}, \lambda\kappa(l_{m-1}h)) = \lambda\kappa(l_{m-1}h) < \kappa(l_{m-1}h)$, which is presented as the situation of Ⅱ in Figure 2. And if $\kappa(l_{m}h)$ is negative, $\kappa(l_{m}h) = \mathop{max}(\kappa_{min}, \lambda\kappa(l_{m-1}h)) = \kappa_{min}$, which is presented as the situation of Ⅲ in Figure 2. In the above three situations, denoted as Ⅰ, Ⅱ, and Ⅲ in Figure 2, a smaller $\kappa(l_mh)$ is set for higher transmission frequency when compared to the latest sampling dynamic parameter $\kappa(l_mh)$. In contrary, when the absolute value of current sampling state is smaller than that of the latest transmitted state larger ($\mid\mid x(l_mh)\mid\mid-\mid\mid x(t_kh)\mid\mid\leq0$), the system tends to be more stable. Then $\xi(l_mh)\leq0$, $\lambda\in (1, 1+\rho)$ and $\lambda\kappa(l_{m-1}h)) > 0$, then $\kappa(l_{m}h)) = min (\kappa_{max}, \lambda\kappa(l_{m-1}h)) = \lambda\kappa(l_{m-1}h) > \kappa(l_{m-1}h)$, which is presented as the situation of Ⅳ in Figure 2. In this case, when compared to the latest sampling dynamic parameter $\kappa(l_{m-1}h)$, the larger $\kappa(l_{m}h)$ is used for a lower communication frequency, saving more communication bandwidth. Noting that when $\kappa(l_{m}h)$ iterates with the fluctuation of every sampling system state, $\kappa_{min}$ and $\kappa_{max}$ are the lower and upper bounds of $\kappa(l_{m}h)$. In summary, compared with the general ETM like ^[23,26,27] with a constant threshold, the threshold can be dynamically adjusted as the fluctuating trend of the current sampling state in every sampling instant in our proposed AETM (2.3).

Remark 2: According to the definition of Eq (2.4), the value of $\kappa(l_mh)$ is limited by $[\kappa_{min}, \kappa_{max}]$. When $\kappa_{min} = \kappa_{max} = 0$, the proposed AETM (2.3) can reduce to traditional time triggering mechanism. When $\kappa_{min}\neq 0$ and $\rho = 0$, the proposed AETM (2.3) will degrade to the general ETM like ^[23,26,27] with a constant threshold. Thus, the proposed AETM is a more general case compared with other published ETMs ^{[23,26,27,38,39,40]}.

2.3.   Sampling-data model of NCSs considering AETM and actuator faults

The transmission process of signals is presented in Figure 3. It can be seen from Figure 3 that the time triggering mechanism is adopted by the sensor with sampling sequence $\mathcal{S}_1 = {0, h, 2h, \dots, kh}, k\in \mathbb{R}$ like ^[38]. When the triggering condition is violated, the AETM transmits the selected sampling data $x(t_kh)$. The triggering transmission sequence set is expressed as $\mathcal{S}_2 = {t_1h, t_2h, \dots, t_kh}, k\in \mathbb{R}$. Considering the packet dropout and packet disorder in the communication network, the sequence set of control signals successfully arrived at the actuator is shown as $\mathcal{S}_3 = {t_1, t_2, \dots, t_k}, k\in \mathbb{R}$. Apparently, there is $\mathcal{S}_3\in\mathcal{S}_2\in \mathcal{S}_1$.

It can be seen from Figure 3, the sampling data $x(t_kh)$ selected to transmission by AETM is transmitted to the controller, which forms the control output. And from Figure 3, the actuators receive control actions from controller as follows:

And the update period of the actuator is related to the trigger period of the AETM, the network communication delay, and the number of packet dropout/disorder, which will lead to the uncertainty of the update period of the actuator. The derivation of the update period of actuator $T_k$ is as follows:

Where $\gamma$ denotes the continuous count of packet dropout/disoder caused by network transmission. Different $k$ leads to different $T_k$, indicating the uncertainty of $T_k$. $T_1$ and $T_2$ denote the minimum and maximum of $T_k$.

Meanwhile, taking the random failures of actuators in NCSs into account, it is assumed that:

There are $m$ independent random variables $f_s(t)(s = 1, 2, \dots, m)$, where $f_s(t)$ is an independent random variable that varies in the range of $[0, \varpi]$. $\varpi$ denotes a constant and $\varpi\geq 1$. The expectation and variance of $f_s(t)$ are assumed to be expressed by $\bar f_s$ and $\delta^2_{fs}$, respectively. Thus, the following definition can be obtained:

$\bar F$ denotes the expectation of $F$.

Defining $\mathcal{U}_l = [t_k, t_{k+1})$, the system delay version can be built for the every two update instants of actuator in front and rear by defining $\tau(t) = t-t_kh$. Then $\dot \tau(t) = 1$, $\tau(t)\in [\tau_m, \tau_M], \tau_M > \tau_m$. $\tau_m = \underline \tau$ denotes the minimum of communication network delay, and $\tau_M = \bar \tau+\bar \gamma+\bar {T}_{kh}$, where $\bar \tau$, $\bar \gamma$, $\bar {T}_{kh}$ denotes the maximum of communication network delay, the maximum of continuous count of packet dropout/disoder, the maximum triggering period of AETM respectively.

Combining (2.1), (2.5), (2.7) and (2.8), the sampling-data model of NCSs considering AETM and actuator faults can be constructed as follows:

2.4.   Control objective

Taking the external disturbances into consideration, an appropriate controller is designed considering actuator failures like Eq (2.7) under the AETM (2.3) while guaranteeing a stable performance. In detail, the following two states should be ensured.

1) The system (2.9) is asymptotically stable with $H_\infty$ performance when there are no external disturbances, $\Delta w(t) = 0$.

2) When $\Delta w(t)\in \mathcal{L}_2[0, \infty)$, there is a positive scalar $\gamma$ such that $\mid\mid y(t)\mid\mid \leq \gamma \mid\mid \Delta w(t) \mid\mid_2$ holds under the zero initial condition.

2.5.   Main lemma

Before deriving the stability criteria, Lemma1 should be introduced.

Lemma 1^[41]: For a given matrix $Z > 0$ and a differentiable function ${x(.):[a, b]\rightarrow \mathbb{R}^n}$, the following inequality holds:

where

and $c_{lv} = \sum\limits_{j = 0}^v\frac{f_{vj}}{l+j+1}/\sum\limits_{j = 0}^v\frac{f_{vj}}{v+j+1}$. The multiple integral terms with degree $k = 1, 2, \dots, r$ are denoted as

Remark 3: Setting $r = 0, r = 1, r = 2$ separately, for example, Eq (2.10) can be transformed into the Jensen inequality ^[42], the Wirtinger-based inequality ^[43], and the auxiliary function based inequality ^[44]. Applying integer $r$ to regulate computation accuracy and complexity synchronously. It can be seen that Eq (2.10) estimates the integral quadratic form $\int^b_a\dot x^T(s)Zx(s)ds$ without conservatism by setting a relatively large $r$. At the same time, more decision variables are introduced inevitably by increasing $r$, resulting in computational complexity. In other word, owing to the adjustability of $r$, the trade-off between computational accuracy and complexity will be obtained in Lemma 1.

3.   $H_\infty$ stability performance analysis and $H_\infty$ reliable controller design

The main results are shown in this section, which are the $H_\infty$ stability condition and the $H_\infty$ reliable event-triggering controller design approach under AETM.

3.1.   $H_\infty$ stability performance analysis

The robust $H_\infty$ performance analysis for NCSs considering actuator failures and AETM is presented in Theorem1.

Theorem 1. The system (2.9) is asymptotically stable with a $H_\infty$ norm bound $\gamma$ for some given parameters $\tau_m, \tau_M$, and $\delta_{fs}(s = 1, \dots, m)$, if there exist positive definite matrices $P$, $Q_1$, $Q_2$, $\Phi$, $R_1$, $R_2$, $Z$, such that the following inequality holds:

where $\Omega_3 = diag[-R_1^{-1}, -R_2^{-1}, -Z^{-1}, -I]$, $\quad\Omega_5 = diag[\widetilde{R_1}, \widetilde{R_2}, \widetilde{Z}], \quad \widetilde{R_1} =$ $diag\{\underbrace{-R_1^{-1}}_m\}, \quad\widetilde{R_2} =$ $diag\{\underbrace{-R_2^{-1}}_m\}, \quad \widetilde{Z} = diag\{\underbrace{-Z^{-1}}_m\}, \quad \Omega_7 = diag\{\underbrace{I}_m\}$.

The element in $\Omega_{10}$ is just $\vartheta_{12.12} = -\gamma^2$, and the elements $\varphi_{ij}(i, j = 1, 2\dots 12)$ in $\Omega_1$ are listed as following:

Proof: At first, the following Lyapunov-KrasovskⅡ functional (LKF) candidates are considered.

where

Considering that the state space model (2.9) of the system contains random variable $F$, we can get the expectation of derivative for $V(t)$ as:

Where $\bar A = Ax(t)+A_d\bar FKx(t-\tau(t))+B\Delta w(t)$. By setting $r = 2$, the Lemma1 can be used to deal with the integral terms in $\dot V_3(t)$, as shown below:

where $\nu_i(i = 0, 1, 2)$ can be obtained from (2.10), and

where

Defining

Considering the AETM condition (2.3), the following inequality can be obtained:

Therefore, considering (2.10)–(3.6) together, the following inequality can be obtained as:

Where $\Omega_1$ has been defined in (3.1) and $\Pi_1 = Ae_1+A_d\bar FKe_3+Be_{12}, \quad e_i = [0_{n\times(i-1)n} I_n 0_{n\times(12-i)n}]$.

Following that, the robust $H_\infty$ performance of the system (2.9) in the presence of external disturbance should be considered. The following inequality can be established:

$\mathbb{E}\{\dot V(t)\}\leq -\mathbb{E}\{\mid\mid y(t)\mid\mid$ can be derived from (3.8) with $\Delta w(t) = 0$. That is to say $\mathbb{E}\{\dot V(t)\} < 0$. The positive definite matrices $P$, $Q_1$, $Q_2$, $\Phi$, $R_1$, $R_2$, $Z$ make $V(t) > 0$. It means that the energy function $V(t)$ containing the system state gradually tends to $0$ with the advance of time. The system (2.9) is asymptotically stable when the external disturbance is $0$ (with $\Delta w(t) = 0$).

Owing to $\mathbb{E}\{\dot V(t)\}$ is continuous in $t$, taking integration both side of (3.8) from $0$ to $+\infty$, we can derive that

Due to the initial condition $\mathbb{E}\{V(0)\} = 0$ and $V(t) > 0$, inequality (Eq 3.9) is equivalent to $\int^{+\infty}_{0}(\mathbb{E}\{-y^T(t)y(t)\}+\gamma^2\Delta w^T(t)\Delta w(t))dt > 0$. Furthermore $\mid\mid y(t)\mid\mid_2\leq\gamma\mid\mid \Delta w(t)\mid\mid_2$ can be obtained. As a result, system (2.9) is asymptotically stable and has a $H_\infty$ performance.

Equation (3.8) can be expressed as:

That is to say $\mathbb{X} < 0$, where:

Thus, the inequality (Eq 3.11) can be transformed into inequality (Eq 3.1) based on Schur complement.

There is $\Pi_2 = Ce_1+D\bar F Ke_3$. And $\Omega_1$, $\Omega_2$, $\Omega_3$, $\Omega_4$, $\Omega_5$, $\Omega_6$, $\Omega_7$, $\Omega_{10}$ have been presented in Theorem 1.

3.2.   $H_\infty$ controller design

While it can be seen from $\varphi_{1.3} = PA_d\bar FK+\frac{\pi^2}{4}Z$ in $\Omega_1$, the multiplication of two unknown variables $K$ and $P$ make matrix inequality (Eq 3.1) nonlinear and the controller gain $K$ cannot be solved by LMI tool of Matlab. The purpose of this paper is to solve the event-triggered reliable $H_\infty$ controller gain $K$. Theorem 2 is an extension of Theorem 1. The nonlinear inequality (Eq 3.1) in Theorem 1 is processed into linear inequality (Eq 3.12) in Theorem 2. Through the LMI tool of MATLAB, the controller gain $K$ can be obtained by Theorem 2.

Theorem 2. The system (2.9) is asymptotically stable with a $H_\infty$ norm bound $\gamma$ for some given parameters $\tau_m, \tau_M, \delta_{fs}(s = 1, \dots, m)$, if there exist positive definite matrices $X$, $\bar Q_1$, $\bar Q_2$, $\bar \Phi$, $\bar R_1$, $\bar R_2$, $\bar Z$, such that the following inequality holds:

where $\bar\Omega_3 = diag\{\bar R_1-2X, \quad \bar R_2-2X, \quad \bar Z-2X, \quad -I\}, \quad \bar\Omega_5 =$ $diag\{\widehat{R_1}, \quad \widehat{R_2}, \quad\widehat{Z} \}, \quad \widehat{R_1} =$ $diag\{\underbrace{\bar R_1-2X}_m\}, \widehat{R_2} = diag\{\underbrace{\bar R_2-2X}_m\}, \widehat{Z} =$ $diag\{\underbrace{\bar Z-2X}_m\}$.

The elements $\varphi_{ij}(i, j = 1, 2\dots 12)$ in $\bar\Omega_1$ are listed as following:

the element in $\Omega_{10}$ is just $\vartheta_{12.12} = -\gamma^2$.

Proof: Define $X = P^{-1}$, \quad $\bar Q_i = XQ_iX, (i = 1, 2)$, \quad $\bar\Phi = X\Phi X$, \quad $\bar R_i = XR_iX(i = 1, 2)$, \quad $\bar Z = XZX$ and $Y = KX$. The inequality (Eq 3.12) can be obtained from pre- and post- multiplying both sides of (3.1) by $diag\{\underbrace{X}_{11}, I, \underbrace{X}_3, I, \underbrace{X}_{3m}, \underbrace{I}_m\}$. The non-linear terms in (3.12) can be estimated by the fact of $-HG^{-1}H\leq G-2H$ for appropriate matrices like ^[25]. And after the above processing, the nonlinear term $PA_d\bar F K$ (with two unknown variables $P$ and $K$) in $\varphi_{1.3}$ of $\Omega_1$ in LMI (3.1) will become $XPA_d\bar F KX = P^{-1}PA_d\bar F KP^{-1} = A_d\bar F KX$. Making $Y = KX$, the nonlinear term $XPA_d\bar F KX$ will become the linear term $A_d\bar F Y$ (with only one unknown variables $Y$). At the same time, $PA$ in LMI (3.1) will become $XPAX = P^{-1}PAP^{-1} = AX$. Therefore, all nonlinear terms become linear terms. Using the LMI toolbox in MATLAB, all unknown variables can be solved, including $Y$ and $X$. Thus $K = YX^{-1}$ will be obtained.

Remark 4: Theorems 1 and 2 are time-varying due to the characteristic of $\kappa(l_mh)$ changing dynamically at every sampling instant. Because the arctangent function atan(.) varies with time. It is a challenge to analyze control system stability and obtain controller gain. Because $\kappa(l_mh)$ varies in $(\kappa_min, \kappa_max)$, when Theorem 1 is used to analyze the $H_\infty$ performance stability and Theorem 2 is used to obtain the controller gain with a given $H_\infty$ performance index, the time-varying $\kappa(l_mh)$ can be substituted with $\kappa_min$ for less conservatism.

4.   Illustrative example

A simulation example is applied to prove the progressiveness of the proposed control method under the AETM in this section.

The pendulum example illustrated in ^[25,38] is listed as follows:

Where $M = 10, \quad m = 1, \quad l = 3, \quad g = 10$ denotes the cart mass, the pendulum bob mass, the length of the pendulum arm and the gravitational acceleration respectively. The initial state is $x(0) = [0.98, \quad 0, \quad 0.2, \quad 0]^T$ as ^[25]. $\gamma = 200, \quad \tau_m = 0.01\quad \kappa_{min} = 0.005, \quad \kappa_{max} = 0.1$ and $\Delta w(t) = 0.01sin(2\pi t)$ are set. And making $F = I$ means the actuator is healthy. The corresponding feedback controller and triggering matrix are achieved as:

It can be seen from Remark 2, if $\kappa_{min}\neq 0, \rho = 0$, the proposed AETM reduces to a general ETM with a constant triggering parameter. Noting that time-triggering is adopted by the sensor in Figure 2 with a trigged period of $h = 0.1$, the sampling period of AETM is $h = 0.1$. On this basis, the average triggering intervals (ATIs) calculated by our proposed AETM when $\rho = 0, \rho = 0.8$ are listed in Table 1 and compared to the results in ^[6,25,38].

Remark 5: It is worth noting that the security control problem of NCSs is investigated by ^[25] under the DoS attack. The ATI of three cases: no DoS attack, a probabilistic DoS attack, and the worst DoS attack are all calculated and shown in ^[25]. Thus, the ATI of ^[25] in Table 1 refers to the case of no DoS attack, which is also considered in this work. Meanwhile, the triggering parameters in ^[25] and ^[38] are all set as constants, which are unaffected by system state fluctuations. However, in AETM (2.3), the triggering parameter $\kappa(l_mh)$ is set as a dynamic adjustment value based on system state fluctuation. When the system state fluctuates severely, the control signal will be transmitted frequently. When the system state is well controlled, the control signal should be reduced dynamically. Additionally, the triggering parameter in ^[6] is set as a dynamic adjustment value rather than a constant. The dynamic adjustment mechanism of ^[6] is different from the method in this work.

When compared to the results in ^[25] and ^[38] in Table 1, our proposed AETM can improve ATI regardless of $\rho = 0$, $\rho = 0.8$. Setting $\rho = 0$, the AETM reduces to $t_{k+1}h = t_kh+\mathop{min}\limits_{m\in\mathbb{N}}\{mh\mid e^T(l_mh)\Phi e(l_mh)\geq \kappa_{min} x^T(t_kh)\Phi x(t_kh)\}$ like the traditional ETM ^[23,26,27]. $\kappa(l_mh)$ is always $\kappa_{min}$, indicating that it always transmits data at a high frequency. Setting $\rho\neq 0$ indicates a faster change rate speed of $\kappa(l_mh)$. $\kappa(l_mh)$ can be dynamically adjusted more flexible as the distance between the current state and the latest transmitted state, resulting in a better average interval. From Table 1, the ATI of $\rho = 0.8$ is larger than $\rho = 0$. Furthermore, even after reducing the proposed AETM to traditional ETM, the ATI is still better than the results in ^[25] and ^[38], because the proposed Theorem 1 derived from Lemma 1 is less conservative than those in ^[25] and ^[38]. Moreover, the setting of the sensitivity parameter $\rho$ in the proposed ATEM (2.3) enables us to dynamically adjust the trigger effect, which is not considered by AETM in ^[6]. It can be seen that when $\rho = 0.8$, the ATI is better than that in ^[6], which illustrates the effect of reducing network bandwidth occupancy is better than the AETM in^[6]. Therefore, compared with the AETM in ^[6], the proposed AETM (2.3) is more flexible.

When the actuator meets a partial failure and $\bar f_1 = 0.95, \quad \delta_{f1}^2 = 0.05$ is set, the system state responses by setting $\rho = 0$, $\rho = 0.8$ are shown in Figures 4 and 5. It can be seen from Figures 4 and 5 that when ETM with $\rho = 0$, or $\rho = 0.8$ is used, even if encountering a certain degree of actuator fault, the system state converges to zero with good performance under our proposed reliable $H_\infty$ control method.

When $\rho = 0$, AETM degrades to normal ETM with a fixed triggering parameter $\kappa_0$. But when $\rho = 0.8$, the triggering parameter $\kappa(l_mh)$ is dynamically adjusted with the fluctuation of the system state. And the change trajectories about $\kappa(l_mh)$ of $\rho = 0.8$ is illustrated in Figure 6. As shown in Figure 6, at the beginning of control, there exists a large error between the front and rear state. The parameter $\kappa(l_mh)$ fluctuates dramatically. When the system state gradually converges to zero, $\kappa(l_mh)$ is reaching to its maximum $\kappa_{max}$, which keeps a low transmitting frequency.

Furthermore, Figures 7 and 8 show the triggering interval with regard to $\rho = 0$, $\rho = 0.8$. It is obvious that the triggering interval of $\rho = 0.8$ is bigger than $\rho = 0$. Therefore, the conclusions in Figures 7 and 8 are consistent with those in Table 1, which illustrates the effect of reducing network bandwidth occupancy of our proposed AETM ($\rho\neq 0$) is better than the traditional ETM ($\rho = 0$).

In this example, a single independent random variable $f_1(t)$ (with $\bar f_1 = 0.95, \quad \delta_{f1}^2 = 0.05$) is taken to illustrate the random failure of the actuator in NCSs, which is assumed that $u^W(t) = f_1(t)u(t)$. That is to say, the control signal $u(t)$ arrives at the actuator, and after the processing of $u^W(t) = f_1(t)u(t)$, the control signal $u(t)$ becomes $u^W(t)$. From the above analysis, we can know $\rho = 0$ (traditional ETM) has a higher signal transmitting frequency than $\rho = 0.8$ (our proposed AETM). That is to say, the control signals $u(t)$ received by the actuator with $\rho = 0$ are more than that with $\rho = 0.8$. When the actuator receives the control signal, it will trigger $f_1(t)$ to generate a random value, which represents the actuator's fault. Thus, the probability of actuator faults triggering ($f_1(t)$) shown in Figure 10 fluctuates more intensively than that in Figure 9.

5.   Conclusions

Considering actuator faults with AETM for NCSs, a novel event-triggered $H_\infty$ reliable control approach is developed in this paper. First, the randomly occurring actuator faults are governed by a series of independent stochastic variables satisfying certain probability distributions. Then, according to the fluctuating trend of the current sampling state, the triggering threshold parameter is adaptively adjusted in the proposed AETM. As a result, the sampling-data model of NCSs considering AETM and actuator faults is constructed. Furthermore, a novel integral inequality is used to analyze the stability performance of $H_\infty$ and derive the event-triggered $H_\infty$ reliable control approach. In the end, case studies are presented to verify the effectiveness of the approach proposed in this paper. However, the security threats in actual communication networks are not considered in this paper, such as denial of service (DoS) attacks and deception attacks. The event-triggered reliable control approach considering various network attacks and actuator faults for NCSs will be the focus of our following work.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant (Nos. 61703060 and 12061088), the Sichuan Science and Technology Program under Grant Nos. 21YYJC0469 and 23ZDYF0645, the Project is funded by China Postdoctoral Science Foundation under Grant Nos. 2020M683274 and 2021T140092, Opening Fund of Geomathematics Key Laboratory of Sichuan Province (scsxdz2018zd04 and scsxdz2020zd01), the Open Research Project of Key Laboratory of Intelligent Manufacturing Quality Big Data Tracing and Analysis of Zhejiang Province, China Jiliang University Grant No. ZNZZSZ-CJLU2022-06.

Conflict of interest

The authors declare there is no conflicts of interest.