1. Introduction
Network control systems have been widely used in many applications such as video surveillance, satellite clusters, offshore platforms, and mobile robotics, among others, because of their advantages of wireless connectivity, efficiency, and flexibility [
1,
2,
3,
4]. However, introducing a network into a control loop can cause some problems, especially when the communication bandwidth is limited. It is indicated that only a few system components can obtain communication resources for data exchange simultaneously, which may affect the system performance and even cause system instability. Furthermore, the main constraint of wireless sensor networks is the limited battery life. Normally, it is impractical to replace batteries so the lifetime of the network control systems is equal to its battery life. The best option to lengthen the battery life is to reduce the wireless communication, which is a major source of energy consumption. The disadvantage of traditional periodic communication and control is that even when the output fluctuation is sufficiently small to change the output signal, the measurement value is also transmitted, resulting in a waste of communication and energy resources of battery-based devices [
5,
6,
7].
The propositions of “replacing periodic control with event-triggered control” have been known since 1950s [
8,
9]. In addition, the interest on event-triggered control was initiated by the paper [
10]. The basic idea of event-triggered control is that communication data based on the measured signals (states or outputs) are sent only when the designed conditions of the event-triggered strategy are satisfied, which can reduce unnecessary calculation and transmission, lower the requirement of a communication network system, and achieve a better balance between the control performance and traffic load [
11]. This is particularly important when multiple systems use a shared network to communicate. Compared with time-triggered systems, shared networks can support more event-triggered systems [
12]. The event-triggered control scheme can save energy resources of battery-based devices, computation resources, and limited network resources as well [
13].
In the last decade, event-triggered control has become a hot research topic and significant contributions have been made [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26]. The event-triggered data sampling strategies based on send-on-delta have been investigated in [
19,
20]. Recent discussions of event-triggered control for stochastic systems could be found in [
21,
22]. Literature [
23,
24] studied the event-triggered strategy of uncertain systems, and some of them are applicable to nonlinear systems. In addition, the event-triggered strategy for transmission time-delayed systems was studied in literature [
25,
26]. However, the reliability of the sensors cannot always be guaranteed because of actuator faults and unknown disturbances, the problem of fault-tolerant control has been actively investigated [
27,
28,
29]. Notably, event-triggered fault-tolerant control for stochastic systems with state delays has not been adequately addressed. Thus, the main contributions of this paper can be summarized as follows: two novel event-triggered fault-tolerant control strategies are proposed based on a state and a send-on-delta event generators for a stochastic system with state delays. The closed-loop networked control system is exponentially mean-square stable, and the prescribed
${H}_{\infty}$ disturbance attenuation performance is also achieved. A simple algorithm is developed to deal with the addressed problem, which can be easily implemented using an efficient linear matrix inequalitie (LMI) toolbox.
This paper is organized as follows.
Section 2 formulates the problem and some important lemmas are presented. Our main results are described in
Section 3, the state-based event-triggered controls are presented in
Section 3.1.
Section 3.2 describes the send-on-delta strategy. Two numerical examples are presented in
Section 4 to illustrate the results.
Section 5 concludes this paper.
Notations: The superscript “T” stands for matrices transport. ${R}^{n}$ and ${R}^{n\times m}$ denote n dimensional Euclidean space and set of all $n\times m$ matrices, respectively. For a square matrix S, $S>0\left(S<0\right)$ means that this matrix is positive definite (negative definite). In symmetric block matrices, “*” is used as an ellipsis for terms induced by symmetry. I denotes an identity matrix with appropriate dimensions. Let $(\mathbf{\Omega},\phantom{\rule{0.277778em}{0ex}}\mathbf{F},\phantom{\rule{0.277778em}{0ex}}\mathbf{P})$ be a complete probability space, where $\mathbf{\Omega}$ is the sample space, $\mathbf{F}$ is the $\sigma $-algebra of subsets of the sample space, and $\mathbf{P}$ is the probability measure on $\mathbf{F}$. Furthermore, $E(\xb7)$ denotes the mathematical expectation of a matrix. $\u2225.\u2225$ stands for standard Euclidean norm in ${R}^{n}$.
2. Problem Statement
Consider the following discrete-time linear stochastic system with state delays defined in a probability space
$(\mathbf{\Omega},\phantom{\rule{0.277778em}{0ex}}\mathbf{F},\phantom{\rule{0.277778em}{0ex}}\mathbf{P})$:
where
k is a discrete-time index,
$x(k)\in {R}^{n}$ is the state vector,
$u(k)\in {R}^{m}$ denotes the control vector, and
$x(0)$ is the initial state.
$z(k)\in {R}^{p}$ correspond to the controlled output variables. The stochastic variable
${w}_{k}$ is a scalar Wiener process defined on a complete space
$(\mathbf{\Omega},\phantom{\rule{0.277778em}{0ex}}\mathbf{F},\phantom{\rule{0.277778em}{0ex}}\mathbf{P})$ with
$E\left({w}_{k}\right)=0$,
$E\left({w}_{k}^{2}\right)=1$ and
$E\left({w}_{i}{w}_{j}\right)=0$ (
$i\ne j$). Moreover, the fault signals
$f(k)$ and disturbance signals
$d(k)$ are assumed to be
${\ell}_{2}$ signals (
$f,w\in {\ell}_{2}^{s}$), where
d is a delay coefficient. The matrices
A,
${A}_{2}$,
${A}_{d}$,
D,
F and
Z are known constant matrices with appropriate dimensions.
This paper assumes controllers and sensors to be collocated or hard-wired. The architecture of the event-triggered network control system used in this study is shown in
Figure 1, which is similar to literature [
7]. The event-triggered mechanism is composed of two units: a feedback controller and a trigger mechanism (event strategy, conditions, or algorithm). The trigger mechanism determines whether the control input should be sent to the actuator via the network. In this event-triggered mechanism, the event condition based on the current controlled output is monitored continuously. Once the condition is satisfied, an event is triggered.
The plant is assumed to be time-driven, whereas the actuator is event-triggered. The actuator is triggered only when a new control vector $u\left(k\right)$ is received. Further, $a\left(k\right)\in \{0,1\}$ is defined as an event-triggered decision variable that determines whether to send the control vector at each sampling time: when $a\left(k\right)=1$, $u\left(k\right)$ can be calculated and sent out; when $a\left(k\right)=0$, $u\left(k\right)$ cannot be allowed to update. In this study, the event-triggered mechanism and feedback controller are co-designed. The objective is to use the minimum trigger time to maintain the control performance.
Remark 1. The design of the event-triggered strategy must specify the minimum trigger time to avoid the zeno phenomenon, i.e., an infinite number of trigger times in finite time [7]. The system event generator used in this study is time-driven and sampled at a constant frequency; thus, the minimum trigger time of an event-triggered strategy is the sampling time, and hence, no zeno phenomenon can occur. Before proceeding further, it is necessary to introduce the definition of mean-square stability.
Definition 1. [30] A discrete stochastic process ${\xi}_{k}$ is said to be exponentially mean-square stable, if there exist constants ${\alpha}_{1}>0$ and $0<{\alpha}_{2}<1$ such that where ${\mathbb{I}}^{+}$ is the set of positive integers and d is a constant. With the help of Definition 1, this paper focuses on the co-design of the feedback controller and the event-triggered mechanism such that the discrete-time linear stochastic system (
1) satisfies the following requirements simultaneously.
- 1:
When $\partial (k)=0$, the system is exponentially mean-square stable.
- 2:
Under the zero-initial condition
for all nonzero
$\partial (k)$, where
$\partial (k)=\left[\begin{array}{c}d(k)\\ f(k)\end{array}\right]$, and a
$\gamma >0$ is prescribed scalar.
Some lemmas are presented, which will play an important role in the proof of our main theorems in
Section 3.
Lemma 1. [31] Suppose $Y>0$, $x\in {R}^{n}$ and $w\left(k\right)$ is a Gaussian random vector satisfying $E\left(w\left(k\right)\right)=0$, $E\left({w}^{2}\left(k\right)\right)=Q$. Let η be the random variable then
$E\left(\eta \right)={x}^{\mathrm{T}}Yx+trace\left({Q}_{}Y\right)$. Lemma 2. (S-procedure [32], Lemma 3) Let $f\left(x\right)$ and $g\left(x\right)$ be two arbitrary quadratic forms over ${R}^{n}$. Then satisfying $g\left(x\right)<0$ if and only if there exist a scalar $\tau \ge 0$ such that $f\left(x\right)-\tau g\left(x\right)\le 0$ for $\forall x\in {R}^{n}$. Lemma 3. [33] Let the matrix $B\in {\mathrm{R}}^{n\times m}$ be of full-column rank with singular value decomposition following the structureif there exist positive–definite matrices $P\in {\mathrm{R}}^{n\times n}$ satisfyingthen there exists an invertible matrix $M\in {\mathrm{R}}^{m\times m}$ such that $PB=BM$, where ${M}^{-1}=V{\Sigma}^{-1}{P}_{11}^{-1}\phantom{\rule{1.0pt}{0ex}}\Sigma {V}^{\mathrm{T}}$.