Dynamic Event-Triggered Control for Sensor–Controller–Actuator Networked Control Systems

Abstract

We consider the problem of output feedback stabilization of LTI systems under event-triggering implementation. In particular, we assume that both the plant output and the control input are both transmitted over the network in an asynchronous manner. To that end, two independent event-triggering rules are constructed to generate the transmission instants of the submitted signals. The proposed approach is dynamic in the sense that the triggering rules involve internal dynamical variables to allow for further reduction in the communication load. Moreover, the inter-transmission times for both sides of the channel are lower bound by enforced dwell times to prevent the occurrence of Zeno phenomena. The problem is challenging due to mutual interactions between the sampling errors of the plant output and the control input, which requires careful handling to ensure closed-loop stability. The triggering mechanisms are designed by emulation as we first ignore the effect of the network and stabilize the plant in continuous-time. Then, the communication constraints are taken into account to derive the triggering conditions such that the stability of the networked control system is preserved. The required conditions are formulated in terms of a linear matrix inequality. The effectiveness of the technique is demonstrated by numerical simulations.

1. Introduction

Networked Control Systems (NCSs) are control systems that combine control theory and communication networks to regulate and manage physical processes, devices, or systems. In NCSs, various components like sensors, actuators, controllers, and the controlled processes are interconnected through a communication network, allowing for remote monitoring and control [1,2,3]. The NCS architectures find applications in various domains, including industrial automation, robotics, autonomous vehicles, smart grids, and environmental monitoring, due to their advantages, such as improved system flexibility, scalability, remote monitoring capabilities, and the potential for cost savings. However, NCSs also pose unique challenges such as sampling, quantization, bit rate constraints and scheduling. The main scope of this study is to develop a stabilizing sampling scheme for NCSs to reduce the utilization of the scarce communication resources while maintaining a desirable closed-loop response [4,5,6,7,8].

One of the attractive ideas for handling the sampling issue over the network is to allow data transmissions over the network only when it is necessary rather than the traditional periodic sampling paradigm. This has led to the appearance of the event-triggered control (ETC) in which the sampling instants are generated using a state-dependant rule [9,10,11,12,13,14,15,16,17,18,19]. This approach can be more efficient in terms of communication and computational resources, as it only sends control updates when needed. This approach has attracted great research interest in different domains of applications and for different classes of systems. Despite the potential of ETC in reducing the amount of communications, the state-dependent sampling can lead to an infinite number of transmissions in a finite time (Zeno phenomenon), which is a challenging problem to handle [20,21]. To prevent this issue, different solutions have been explored in the literature including ETC with fixed threshold, e.g., [22,23], enforcing a minimum time between transmissions, e.g., [24,25] and periodic ETC, e.g., [26,27,28,29,30,31,32,33]. We are particularly interested in the second approach since it allows for combining time-triggered control with event-triggering rules in an intuitive manner and enables the guarantee of global asymptotic stability for the closed-loop system, which cannot be achieved with the approach of a fixed threshold and is very challenging to ensure with periodic ETC.

We consider the scenario where both the plant output and the control input are sent over the channel using two independent ETC mechanisms. As such, we develop stabilizing dynamic ETCs based on locally available information at each side of the channel. A major challenge in this setup is to handle the mutual interaction between the sampling-induced error of the output and input signals. Moreover, achieving good control performance with asynchronous sampling often involves trade-offs between data transmissions and control updates. Also, if control updates occur too infrequently or rapidly, it can disrupt the system’s stability, causing oscillations or even instability. Hence, the modeling and the stability analysis of the considered setup requir careful investigation to ensure overall stability while reducing the sampling frequency over the network [34,35]. The proposed ETC strategy is dynamic in the sense that the triggering rule involves a dynamic variable, which has the potential to further reduce the number of transmissions compared to static triggering rules; see, e.g., [36,37,38,39]. The ETC is designed by using the emulation approach, where an observer-based controller is synthesized first in continuous time to stabilize the plant. Then, the communication constraints are introduced and we construct the triggering rules to preserve the closed-loop stability. The overall system is modeled as a hybrid dynamical system to truly describe the mixed continuous-time and discrete-time behaviors that naturally arise in sampled-data systems. The required conditions are expressed in terms of a linear matrix inequality (LMI), which provides a systematic design procedure for the proposed approach. The effectiveness of the technique has been illustrated by simulation.

The main contribution of this study is summarized below.

• We develop asynchronous dynamic ETC mechanisms for output feedback LTI systems that ensure the global asymptotic stability of the closed-loop system and reduce the data rate over the network.
• The overall problem is modeled as a hybrid dynamical system to account for the existing dynamical phenomena.
• The design methodology is presented systemically by solving an LMI condition to facilitate the application of the proposed approach.

The rest of the paper is organized as follows. Preliminaries are given in Section 3. The problem is formally stated in Section 4. The stability result is given in Section 7. The simulation result is presented in Section 8. Conclusions are provided in Section 9.

2. Related Work

The problem of asynchronous event-triggered control for both the plant output and the control input for continuous-time systems has been considered in a few works of literature [40,41,42,43] to the best of our knowledge. The approach of [40] is developed for networked control systems with decentralized event-triggering of the output measurements over the network. Moreover, to further reduce the communication load, the feedback measurement is sent to the controller using a single bit. The approach ensures uniform global practical stabilization of the closed-loop system. We note that the result of [40] only takes into account the transmission of the output measurement over the network while the control input is assumed to be directly sent to the plant, which is not the case in our study. The authors of [41] proposed an event-triggering mechanism for T-S fuzzy systems under asynchronous network communications. In particular, it is assumed that both the control input and the plant output are sent over the network asynchronously. Then, a global asymptotic stability property is derived for the overall system under a set of LMI conditions. Compared to [41], we address a different class of systems, and our proposed ETC technique is dynamic and not static as in [41], which has the potential to achieve further reduction in the number of communications. Moreover, we analyze the overall system as a hybrid dynamical system, which provides more insights into closed-loop behavior. The result of [42] is dedicated to sensor–controller–actuator networks with physically distributed nodes. Then, the nodes in the network are grouped into three functional layers: the sensor layer, the controller/observer layer, and the actuator layer. Moreover, the flow of information is only from the sensor to the observer to the actuator layer, with the only intra-layer communication occurring in the observer layer. The approach guarantees global asymptotic stability of the origin while ensuring the existence of a positive minimum inter-transmission time for each node. In contrast to [42], our ETC mechanism is dynamic, not static, and the overall system is modeled as a hybrid dynamical system. In [43], an observer-based event-triggering mechanism has been proposed for LTI systems, considering that both the plant output and the control input are transmitted over the network. However, the approach of [43] assumes that the transmission instants for the output measurement and the control input are synchronized using a single triggering rule. Moreover, the proposed ETC mechanism in [43] is static and not dynamic as we consider.

Table 1. Novelty of the proposed approach with respect to existing results.

	[40]	[41]	[42]	[43]	Proposed Approach
Plant Dynamics	nonlinear	T-S Fuzzy	LTI	LTI	LTI
Controller	state feedback	output feedback	output feedback	output feedback	output feedback
Modeling	continuous	continuous	continuous	continuous	hybrid
ETC mechanism	static	static	static	static	dynamic
Stability	practical	asymptotic	asymptotic	asymptotic	asymptotic
ETC performance compared to periodic	–	–	–	–	better performance ensured

highlights the contribution of the proposed approach with respect to existing techniques.

It is worth mentioning that there exist other techniques in the literature for asynchronous event-triggered control, e.g., [44,45,46,47,48]. However, those results are concerned with different classes of systems and rely on a different set of assumptions. Moreover, the dynamic model and the stability analysis are different from our approach.

3. Preliminaries

Let $\mathbb{R}:=(-\infty ,\infty )$, ${\mathbb{R}}_{\ge 0}:=[0,\infty )$ and $\mathbb{N}:=\{0,1,2,\dots \}$. A continuous function $\gamma :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is of class $\mathcal{K}$ if it is zero at zero, strictly increasing, and it is of class ${\mathcal{K}}_{\infty }$ if in addition $\gamma \left(s\right)\to \infty$ as $s\to \infty$. A continuous function $\gamma :{\mathbb{R}}_{\ge 0}\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is of class $\mathcal{KL}$ if for each $t\in {\mathbb{R}}_{\ge 0}$, $\gamma (.,t)$ is of class $\mathcal{K}$ and, for each $s\in {\mathbb{R}}_{\ge 0}$, $\gamma (s,.)$ is decreasing to zero. We denote the minimum and maximum eigenvalues of the real symmetric matrix A as ${\lambda }_{min}\left(A\right)$ and ${\lambda }_{max}\left(A\right)$, respectively. We write $A^T$ to denote the transpose of A, and ${\mathbb{I}}_n$ stands for the identity matrix of dimension n. We write $(x,y)$ to represent the vector ${[x^T,y^T]}^T$ for $x\in {\mathbb{R}}^n$ and $y\in {\mathbb{R}}^m$. For a vector $x\in {\mathbb{R}}^n$, we denote by $\left|x\right|:=\sqrt{x^{T}x}$ its Euclidean norm and for a matrix $A\in {\mathbb{R}}^{n\times m}$, $\left|A\right|:=\sqrt{{\lambda }_{max}\left(A^{T}A\right)}$. We consider hybrid systems of the following form using the formalism of [49]

$$\dot{x}=F\left(x\right)x\in C,x^{+}\in G\left(x\right)x\in D,$$

(1)

where $x\in {\mathbb{R}}^n$ is the state, F is the flow map, C is the flow set, G is the jump map, and D is the jump set. Solutions to system (1) are defined on so-called hybrid time domains. A set $E\subset {\mathbb{R}}_{\ge 0}\times$ is called a compact hybrid time domain if ${\displaystyle E=\bigcup _{j\in \{0,\dots ,J-1\}}([t_j,t_{j+1}],j)}$ for some finite sequence of times $0=t_0\le t_1\le \cdots \le t_J$ and it is a hybrid time domain if for all $(T,J)\in E,E\cap \left(\right[0,T]\times \{0,1,\dots ,J\left\}\right)$ is a compact hybrid time domain.

4. Problem Statement

Consider the LTI plant model

$$\begin{array}{ccc}\hfill \dot{x}\left(t\right)& =\hfill & Ax\left(t\right)+Bu\left(t\right)\hfill \\ \hfill y\left(t\right)& =\hfill & Cx\left(t\right),\hfill \end{array}$$

(2)

where $x\left(t\right)\in {\mathbb{R}}^{n_x}$, $u\left(t\right)\in {\mathbb{R}}^{n_u}$, $y\left(t\right)\in {\mathbb{R}}^{n_y}$ and $A,B,C$ are matrices of appropriate dimensions. Assume that the plant is stabilized by an observer-based controller of the form

$$\begin{array}{ccc}\hfill \dot{\widehat{x}}\left(t\right)& =\hfill & A\widehat{x}\left(t\right)+Bu\left(t\right)+F(y\left(t\right)-C\widehat{x}\left(t\right))\hfill \\ \hfill u\left(t\right)& =\hfill & -K\widehat{x}\left(t\right),\hfill \end{array}$$

(3)

where $\widehat{x}\left(t\right)\in {\mathbb{R}}^{n_x}$ is the estimated state and K, F are designed such that $A-FC$ and $A-KB$ are Hurwitz (this is always possible when plant (2) is stabilizable and detectable).

We study the scenario where the communications between plant (2) and controller (3) are carried out via a digital channel (see Figure 1). In particular, the transmissions of the plant output y to controller (3) and the transmissions of the control input u to plant (2) occur at a priori different sampling instants. We, respectively, denote the sequences of transmissions of y and of u by $t_k^y,i\in \mathbb{N}$, and $t_k^u,k\in \mathbb{N}$. These sequences are defined by two independent triggering conditions, which we design in the following.

At each transmission instant $t_k^y,k\in \mathbb{N}$, the output is sampled and transmitted to the controller. On the other hand, the control input is only broadcasted to the actuators at transmission instants $t_k^u,k\in \mathbb{N}$. We ignore the possible transmission delays in this study.

$$\begin{array}{cccc}\hfill \dot{x}\left(t\right)& =& Ax\left(t\right)+Bu\left(t_k^u\right)\hfill & \forall t\in [t_k^u,t_{k+1}^u]\hfill \\ \hfill \dot{\widehat{x}}\left(t\right)& =& A\widehat{x}\left(t\right)+Bu\left(t\right)+F(y\left(t_k^y\right)-C\widehat{x}\left(t\right))\hfill & \forall t\in [t_k^y,t_{k+1}^y]\hfill \\ \hfill y\left(t\right)& =& Cx\left(t\right)\hfill \\ \hfill u\left(t\right)& =& -K\widehat{x}\left(t\right)\hfill \\ \hfill \dot{y}\left(t_k^y\right)& =& 0\hfill & \forall t\in [t_k^y,t_{k+1}^y]\hfill \\ \hfill \dot{u}\left(t_k^u\right)& =& 0\hfill & \forall t\in [t_k^u,t_{k+1}^u]\hfill \\ \hfill y\left(t_k^{y^{+}}\right)& =& y\left(t_k^y\right)\hfill \\ \hfill u\left(t_k^{u^{+}}\right)& =& u\left(t_k^u\right),\hfill \end{array}$$

(4)

At each transmission instant $t_{k+1}^y$, the last transmitted value of $y\left(t_k^y\right)$ is reset to the actual value of $y\left(t_{k+1}^y\right)$ and then kept constant until the next sampling time by means of ZOH implementation. Similarly, $u\left(t_k^u\right)$ is reset to the actual value of $u\left(t_{k+1}^u\right)$ at $t_{k+1}^u$ and then remains constant until the next update instant by using ZOH implementation. We define the sampling-induced error as

$$\begin{array}{cccc}\hfill e_y\left(t\right)& :=\hfill & y\left(t_k^y\right)-y\left(t\right)\hfill & \forall t\in [t_k^y,t_{k+1}^y]\hfill \\ \hfill e_u\left(t\right)& :=\hfill & u\left(t_k^u\right)-u\left(t\right)\hfill & \forall t\in [t_k^u,t_{k+1}^u],\hfill \end{array}$$

(5)

which are reset to 0 at each corresponding transmission instant. Consequently, we have that for all $t\in [t_k^u,t_{k+1}^u]$

$$\begin{array}{ccc}\hfill \dot{x}\left(t\right)& =\hfill & Ax\left(t\right)+B(u\left(t\right)+e_u\left(t\right))\hfill \\ \hfill & =\hfill & Ax\left(t\right)-BK\widehat{x}\left(t\right)+B{e}_u\left(t\right)\hfill \end{array}$$

(6)

and for all $t\in [t_k^y,t_{k+1}^y]$

$$\begin{array}{ccc}\hfill \dot{\widehat{x}}\left(t\right)& =\hfill & A\widehat{x}\left(t\right)+Bu\left(t\right)+Fy\left(t_k^y\right)-FC\widehat{x}\left(t\right)\hfill \\ \hfill & =\hfill & A\widehat{x}\left(t\right)-BK\widehat{x}\left(t\right)+F(y\left(t\right)+e_y\left(t\right))\hfill \\ \hfill & =\hfill & (A-BK-FC)\widehat{x}\left(t\right)+FCx\left(t\right)+F{e}_y\left(t\right).\hfill \end{array}$$

(7)

Similarly, in view of (5), we obtain

$$\begin{array}{ccc}\hfill {\dot{e}}_y\left(t\right)& =\hfill & -C\dot{x}\hfill \\ \hfill & =\hfill & -CAx\left(t\right)+CBK\widehat{x}\left(t\right)-CB{e}_u\left(t\right)\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill {\dot{e}}_u\left(t\right)& =\hfill & K\dot{\widehat{x}}\left(t\right)\hfill \\ \hfill & =\hfill & K(A-BK-FC)\widehat{x}\left(t\right)+KFCx\left(t\right)+KF{e}_y\left(t\right)\hfill \end{array}$$

(9)

The proposed dynamic event-triggering mechanisms for the output measurements $y\left(t\right)$ and for the control input $u\left(t\right)$ are given by

$$\begin{array}{c}\hfill t_{k+1}^y=inf\left\{t>t_k^y+T_y:{\eta }_y=0\right\}\end{array}$$

(10)

and

$$\begin{array}{c}\hfill t_{k+1}^u=inf\left\{t>t_k^u+T_u:{\eta }_u=0\right\},\end{array}$$

(11)

where $T_y,T_u>0$ are strictly positive constants that we enforce on the inter-transmission times $t_k^y$ and $t_k^u$, respectively. These constants will be designed as the maximally allowable transmission interval (MATI) bounds for time-triggered controller [50] as will be explained later. The variables ${\eta }_y\left(t\right),{\eta }_u\left(t\right)$ are dynamic variables, which evolve according to the following impulsive dynamics

$$\begin{array}{cccc}\hfill {\dot{\eta }}_y& =\hfill & {\Psi }_y(y,e_y,{\eta }_y)\hfill & {\tau }_y\in [t_k^y,t_{k+1}^y)\hfill \\ \hfill {\eta }_y\left(t_k^{y^{+}}\right)& =\hfill & {\eta }_y\left(t_k^y\right)\hfill & {\tau }_y=t_k^y,\hfill \end{array}$$

(12)

and

$$\begin{array}{cccc}\hfill {\dot{\eta }}_u& =\hfill & {\Psi }_u(u,e_u,{\eta }_u)\hfill & {\tau }_u\in [t_k^u,t_{k+1}^u)\hfill \\ \hfill {\eta }_u\left(t_k^{u^{+}}\right)& =\hfill & {\eta }_y\left(t_k^u\right)\hfill & {\tau }_u=t_k^u,\hfill \end{array}$$

(13)

where the functions ${\Psi }_y(y,e_y,{\eta }_y)$ and ${\Psi }_u(u,e_u,{\eta }_u)$ will be specified later. Note that the event-triggering mechanisms (10), (12) and (11), (13) depend only on locally available information at each side of the channel. Our objective is to properly design the dwell times $T_y,T_u$ and the functions ${\Psi }_y(y,e_y,{\eta }_y)$ and ${\Psi }_u(u,e_u,{\eta }_u)$ such that the closed-loop stability is guaranteed. In the next section, we first derive the overall hybrid model and then we proceed to explain how to construct those event-triggering rules.

5. Hybrid Dynamical Model

As shown in the previous section, the implementation of event-triggered control involves interaction between continuous-time dynamics and discrete changes due to the effect of sampling. Hence, it is more intuitive to describe the closed-loop system as a hybrid dynamical system to account for such mixed dynamics. We explain in this section how to develop such a model using the framework of [49] (see also [51,52,53]).

Define $\xi \left(t\right):=(x\left(t\right),\widehat{x}\left(t\right))\in {\mathbb{R}}^{2n_x}$. Then, we obtain

$$\begin{array}{ccc}\hfill \dot{\xi }\left(t\right)& =\hfill & \left[\begin{array}{cc}A& -BK\\ FC& A-BK-FC\end{array}\right]\xi \left(t\right)+\left[\begin{array}{c}0\\ F\end{array}\right]e_y\left(t\right)+\left[\begin{array}{c}B\\ 0\end{array}\right]e_u\left(t\right)\hfill \\ \hfill & =:\hfill & {\mathcal{A}}_1\xi \left(t\right)+{\mathcal{B}}_1{e}_y\left(t\right)+{\mathcal{M}}_1{e}_u\left(t\right).\hfill \end{array}$$

(14)

Similarly, we obtain

$$\begin{array}{ccc}\hfill {\dot{e}}_y\left(t\right)& =\hfill & {\mathcal{A}}_2\xi \left(t\right)+{\mathcal{M}}_2{e}_u\left(t\right)\hfill \\ \hfill {\dot{e}}_u\left(t\right)& =\hfill & {\mathcal{A}}_3\xi \left(t\right)+{\mathcal{B}}_3{e}_y\left(t\right)\hfill \end{array}$$

(15)

with

$$\begin{array}{ccc}\hfill {\mathcal{A}}_2& =\hfill & \left[\begin{array}{cc}-CA& CBK\end{array}\right]\hfill \\ \hfill {\mathcal{B}}_2& =\hfill & 0\hfill \\ \hfill {\mathcal{M}}_2& =\hfill & -CB,\hfill \\ \hfill {\mathcal{A}}_3& =\hfill & \left[\begin{array}{cc}KFC& K(A-BK-FC)\end{array}\right]\hfill \\ \hfill {\mathcal{B}}_3& =\hfill & KF\hfill \\ \hfill {\mathcal{M}}_3& =\hfill & 0\hfill \end{array}$$

(16)

We introduce two timers ${\tau }_y,{\tau }_u\in {\mathbb{R}}_{\ge 0}$ to describe the time elapsed since the last transmissions of y and of u, respectively, which have the following dynamics

$$\begin{array}{ccccccc}\hfill {\dot{\tau }}_y& =\hfill & 1\hfill & \forall t\in [t_k^y,t_{k+1}^y],\hfill & {\tau }_y\left(t_k^{y^{+}}\right)\hfill & =\hfill & 0\hfill \\ \hfill {\dot{\tau }}_u& =\hfill & 1\hfill & \forall t\in [t_k^u,t_{k+1}^u],\hfill & {\tau }_u\left(t_k^{u^{+}}\right)\hfill & =\hfill & 0.\hfill \end{array}$$

(17)

These variables will be useful to define the triggering conditions.

Let $q:=(\xi ,e,\eta ,\tau )\in {\mathbb{R}}^{n_q}$, where $\xi :=(x,\widehat{x})\in {\mathbb{R}}^{n_x}$, $e:=(e_y,e_u)\in {\mathbb{R}}^{n_e}$, $\eta :=({\eta }_y,{\eta }_u)\in \mathbb{R}\times \mathbb{R}$ and $\tau :=({\tau }_y,{\tau }_u)\in \mathbb{R}\times \mathbb{R}$. We model the event-triggered controlled system using the hybrid formalism of [49] (like in [14,54,55]), for which a jump corresponds to a transmission. Hence, we obtain

$$\begin{array}{cccccc}\hfill \dot{q}\left(t\right)& \in \hfill & \mathcal{F}\left(q\right),\hfill & q\left(t\right)\hfill & \in \hfill & \mathcal{C}\hfill \\ \hfill q^{+}\left(t\right)& \in \hfill & \mathcal{G}\left(q\right),\hfill & q\left(t\right)\hfill & \in \hfill & \mathcal{D}\hfill \end{array}$$

(18)

where $\mathcal{C}:={\mathcal{C}}_y\cap {\mathcal{C}}_u$ and $\mathcal{D}:={\mathcal{D}}_y\cup {\mathcal{D}}_u$ with the flow sets ${\mathcal{C}}_y,{\mathcal{C}}_u$ and the jump sets ${\mathcal{D}}_y,{\mathcal{D}}_u$ to be defined later. The flow map is given by

$$\begin{array}{c}\hfill \mathcal{F}\left(q\right)=\left(\begin{array}{c}{\mathcal{A}}_{1}x\left(t\right)+{\mathcal{B}}_1{e}_y\left(t\right)+{\mathcal{M}}_1{e}_u\left(t\right)\\ {\mathcal{A}}_{2}x\left(t\right)+{\mathcal{B}}_2{e}_y\left(t\right)\\ {\mathcal{A}}_{3}x\left(t\right)+{\mathcal{M}}_3{e}_u\left(t\right)\\ {\Psi }_y(y,e_y,{\eta }_y)\\ {\Psi }_u(u,e_u,{\eta }_u)\\ 1\\ 1\end{array}\right),q\left(t\right)\in \mathcal{C}\end{array}$$

(19)

and the jump map is given by

$$\begin{array}{ccc}\hfill \mathcal{G}\left(q\right)& =\hfill & \left\{\begin{array}{cccc}{G}_1\left(q\right),\hfill & q\left(t\right)\hfill & \in \hfill & {\mathcal{D}}_y\setminus {\mathcal{D}}_u\hfill \\ G_2\left(q\right),\hfill & q\left(t\right)\hfill & \in \hfill & {\mathcal{D}}_u\setminus {\mathcal{D}}_y\hfill \\ \left\{G_1\left(q\right),G_2\left(q\right)\right\},\hfill & q\left(t\right)\hfill & \in \hfill & {\mathcal{D}}_y\cap {\mathcal{D}}_u,\hfill \end{array}\right.\hfill \end{array}$$

(20)

with

$$\begin{array}{c}\hfill G_1\left(q\right)=\left(\begin{array}{c}x\\ 0\\ e_u\\ {\eta }_y\\ {\eta }_u\\ 0\\ 1\end{array}\right),G_2\left(q\right)=\left(\begin{array}{c}x\\ e_y\\ 0\\ {\eta }_y\\ {\eta }_u\\ 1\\ 0\end{array}\right)\end{array}$$

(21)

Our main objective is to synthesize asynchronous event-triggering rules, i.e., the sets $C_y$, $D_y$ and $C_u$, $D_u$, to ensure a global asymptotic stability property for system (18) as well as the existence of a strictly positive minimum time between two transmissions of each triggering condition.

$$\left[\begin{array}{ccc}{\mathcal{A}}_1^{T}P+P{\mathcal{A}}_1+\epsilon {\mathbb{I}}_{n_{\xi }}+{\mathcal{A}}_2^T{\mathcal{A}}_2+{\mathcal{A}}_3^T{\mathcal{A}}_3+{\delta }_y{C}_y^T{C}_y+{\delta }_u{C}_u^T{C}_u& P{\mathcal{B}}_1& P{\mathcal{M}}_1\\ {\mathcal{B}}_1^{T}P& -{\tilde{\gamma }}_y{\mathbb{I}}_{n_y}& 0\\ {\mathcal{M}}_1^{T}P& 0& -{\tilde{\gamma }}_u{\mathbb{I}}_{n_u}\end{array}\right]\le 0,$$

(22)

where $C_y:=\left[C0\right]$ and $C_u:=[0-K]$.

6. Design of the Event-Triggering Mechanism

We first present the following technical lemma and then explain how to exploit it to construct the event-triggering mechanism.

Lemma 1. 

Consider system (18). Suppose there exist $\epsilon ,{\delta }_y,{\delta }_u,{\tilde{\gamma }}_y,{\tilde{\gamma }}_u>0$ and a positive definite symmetric real matrix P such that the LMI condition in (22) holds, then the Lyapunov function $V\left(x\right)={\xi }^{T}P\xi$ satisfies

$$\begin{array}{ccc}\hfill \langle \nabla V\left(\xi \right),{\mathcal{A}}_1\xi \left(t\right)+{\mathcal{B}}_1{e}_y\left(t\right)+{\mathcal{M}}_1{e}_u\left(t\right)\rangle & \le \hfill & -{\epsilon \left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-|{\mathcal{A}}_3{x|}^2-{\delta }_y{\left|y\right|}^2-{\delta }_u{\left|u\right|}^2\hfill \\ \hfill & \hfill & +{\tilde{\gamma }}_y|e_y{|}^2+{\tilde{\gamma }}_u{\left|e_u\right|}^2.\hfill \end{array}$$

(23)

The proof of Lemma 1 can be shown straightforwardly by pre- and post-multiplying the LMI condition (22) by the state vector ${(\xi ,e_y,e_u)}^T$ and its transpose, respectively. The conclusion of Lemma 1 means that the closed-loop system achieves an ${\mathcal{L}}_2$-gain property from the input $(e_y,e_u)$ to the state $\xi$, which we use to derive the sampling period and the event-triggering mechanism.

Define $W_y\left(e_y\right)=\left|e_y\right|$ and $W_y\left(e_u\right)=\left|e_u\right|$. Then, in view of (15), it holds that

$$\begin{array}{ccc}\hfill \langle \nabla W_y\left(e_y\right),{\mathcal{A}}_{2}x\left(t\right)+{\mathcal{M}}_2{e}_y\left(t\right)\rangle & \le \hfill & L_y{W}_u\left(e_u\right)+\left|{\mathcal{A}}_{2}x\left(t\right)\right|\hfill \\ \hfill \langle \nabla W_u\left(e_u\right),{\mathcal{A}}_{3}x\left(t\right)+{\mathcal{B}}_3{e}_u\left(t\right)\rangle & \le \hfill & L_u{W}_y\left(e_y\right)+\left|{\mathcal{A}}_{3}x\left(t\right)\right|,\hfill \end{array}$$

(24)

where $L_y:=\left|{\mathcal{M}}_2\right|$ and $L_u:=\left|{\mathcal{B}}_3\right|$. It is important to note, in view of (24), that the dynamics of the sampling error $e_y$ of the output measurement y is affected by the sampling error $e_u$ of the control input u and vice versa. Such mutual interaction between the sampling errors is challenging and needs to be carefully handled to endure closed-loop stability. Based on (23) and (24), the MATI bounds $T_y$ and $T_u$ in [50] can be generalized to our case as follows

$$\begin{array}{ccc}\hfill T_y\left({\gamma }_y\right)& :=\hfill & {\displaystyle \frac{1}{{\gamma }_y}}{\displaystyle \frac{\pi }{2}}\hfill \\ \hfill T_u\left({\gamma }_u\right)& :=\hfill & {\displaystyle \frac{1}{{\gamma }_u}}{\displaystyle \frac{\pi }{2}},\hfill \end{array}$$

(25)

where ${\gamma }_y={\sqrt{\tilde{\gamma }}}_y$, ${\gamma }_u={\sqrt{\tilde{\gamma }}}_u$ with ${\tilde{\gamma }}_y,{\tilde{\gamma }}_u$ come from Lemma 1. To provide some insights on the intuition behind those bounds in (25), let us define ${\varphi }_y,{\varphi }_u:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ to be the solutions to the following differential systems

$$\begin{array}{ccc}{\displaystyle \frac{d{\varphi }_y}{d{\tau }_y}}\hfill & =\hfill & \left\{\begin{array}{cc}-(1+{\kappa }_y){\overline{\gamma }}_y({\varphi }_y^2+1)\hfill & {\tau }_y\in [0,T_y]\hfill \\ 0\hfill & {\tau }_y>T_y,\hfill \end{array}\right.\hfill \\ {\displaystyle \frac{d{\varphi }_u}{d{\tau }_u}}\hfill & =\hfill & \left\{\begin{array}{cc}-(1+{\kappa }_u){\overline{\gamma }}_u({\varphi }_u^2+1)\hfill & {\tau }_u\in [0,T_u]\hfill \\ 0\hfill & {\tau }_u>T_u,\hfill \end{array}\right.\hfill \end{array}$$

(26)

where ${\overline{\gamma }}_y:=\sqrt{{\displaystyle \frac{1}{(1+{\kappa }_y)}}({\gamma }_y^2+{\displaystyle \frac{L_u^2}{{\kappa }_u}}+{ϵ}_y)}$ and ${\overline{\gamma }}_u:=\sqrt{{\displaystyle \frac{1}{(1+{\kappa }_u)}}({\gamma }_u^2+{\displaystyle \frac{L_y^2}{{\kappa }_y}}+{ϵ}_u)}$ for any ${\kappa }_y,{\kappa }_u,{ϵ}_y,{ϵ}_u>0$, and ${\lambda }_y,{\lambda }_u\in (0,1)$. Hence, the time $T_y\left({\gamma }_y\right)$ corresponds to the times it takes for ${\varphi }_y$ to decrease from ${\varphi }_y\left(0\right)={\lambda }_y^{-1}$ to ${\varphi }_y\left(T_y\right)={\lambda }_y$. Similarly, the time $T_u\left({\gamma }_u\right)$ corresponds to the times it takes for ${\varphi }_u$ to decrease from ${\varphi }_u\left(0\right)={\lambda }_u^{-1}$ to ${\varphi }_u\left(T_u\right)={\lambda }_u$.

The functions ${\Psi }_y(y,e_y,{\eta }_y)$ and ${\Psi }_u(u,e_u,{\eta }_u)$ for the event-triggering mechanisms in (10), (12) and (11), (13) are designed as follows

$$\begin{array}{ccc}\hfill {\Psi }_y(y,e_y,{\eta }_y)& :=\hfill & \left\{\begin{array}{cccc}{\delta }_y{\left|y\right|}^2-{\alpha }_y{\eta }_y,\hfill & \hfill & \hfill & {\tau }_y\in [0,T_y]\hfill \\ {\delta }_y{\left|y\right|}^2-{\mu }_y{\left|e_y\right|}^2-{\alpha }_y{\eta }_y,\hfill & \hfill & \hfill & {\tau }_y>T_y,\hfill \end{array}\right.\hfill \end{array}$$

(27)

and

$$\begin{array}{ccc}\hfill {\Psi }_u(u,e_u,{\eta }_u)& :=\hfill & \left\{\begin{array}{cccc}{\delta }_u{\left|u\right|}^2-{\alpha }_u{\eta }_u,\hfill & \hfill & \hfill & {\tau }_u\in [0,T_u]\hfill \\ {\delta }_u{\left|u\right|}^2-{\mu }_y{\left|e_u\right|}^2-{\alpha }_u{\eta }_u,\hfill & \hfill & \hfill & {\tau }_u>T_u,\hfill \end{array}\right.\hfill \end{array}$$

(28)

where ${\mu }_y:={\gamma }_y^2+{\displaystyle \frac{L_u^2}{{\kappa }_u}}+(1+{\kappa }_y){\overline{\gamma }}_y^2{\lambda }_y^2+{ϵ}_y$, ${\mu }_u:={\gamma }_u^2+{\displaystyle \frac{L_y^2}{{\kappa }_y}}+(1+{\kappa }_u){\overline{\gamma }}_u^2{\lambda }_u^2+{ϵ}_u$ with ${\gamma }_y,{\gamma }_u,{\delta }_y,{\delta }_u>0$ from Lemma 1, $Ly,Lu$ comes from (24), ${\alpha }_y,{\alpha }_u>0$ are arbitrary small and ${\kappa }_y,{\kappa }_u$ are defined in (26).

The flow and the jump sets in (18) are now given by

$$\begin{array}{ccc}{C}_y\hfill & :=\hfill & \left\{q\in {\mathbb{R}}^{n_q}:{\tau }_y\le T_y\vee {\eta }_y\ge 0\right\}\hfill \\ D_y\hfill & :=\hfill & \left\{q\in {\mathbb{R}}^{n_q}:{\tau }_y\ge T_y\wedge {\eta }_y\le 0\right\}\hfill \\ C_u\hfill & :=\hfill & \left\{q\in {\mathbb{R}}^{n_q}:{\tau }_u\le T_u\vee {\eta }_u\ge 0\right\}\hfill \\ D_u\hfill & :=\hfill & \left\{q\in {\mathbb{R}}^{n_q}:{\tau }_u\ge T_u\wedge {\eta }_u\le 0\right\}.\hfill \end{array}$$

(29)

Hence, the hybrid dynamical model has been completely defined by (18) and (29). We proceed next to the main result of this study.

7. Stability Result

The following theorem presents the ensured global asymptotic stability property for the system (18) and (29).

Theorem 1. 

Consider system (18) with the flow and the jump sets in (29) and the constant $T_y,T_u$ are designed as in (25). Suppose there exist $\epsilon ,{\delta }_y,{\delta }_u,{\tilde{\gamma }}_y,{\tilde{\gamma }}_u>0$ and a positive definite symmetric real matrix P such that the LMI condition in (22) holds. There exists $\beta \in \mathcal{KL}$ such that the following property is satisfied, for all $(t,j)\in \mathit{dom}q$

$$\begin{array}{c}\hfill \left|q\right(t,j\left)\right|\le \beta \left(\right|q(0,0)|,t+j).\end{array}$$

(30)

Moreover, Zeno behaviors are ruled out for both the plant output and the control input.

Theorem 1 shows that the states of the plant and of the controller, i.e., x, asymptotically converge to the origin. Note that the existence of strictly positive lower bounds on the inter-transmission times of the output measurements and of the control input directly follows from (29).

Proof of Theorem 1. 

Recall that $\mathcal{C}={\mathcal{C}}_y\cap {\mathcal{C}}_u$ and $\mathcal{D}={\mathcal{D}}_y\cup {\mathcal{D}}_u$. We define for all $q\in \mathcal{C}\cup \mathcal{D}$

$$\begin{array}{c}R\left(q\right):=V\left(\xi \right)+{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y\right)|e_y{|}^2+{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u\right){\left|e_u\right|}^2+{\eta }_y+{\eta }_u,\hfill \end{array}$$

(31)

where $V\left(\xi \right):={\xi }^{T}P\xi$ for all $x\in {\mathbb{R}}^{n_{\xi }}$, where P is from Lemma 1.

Dynamics of $R\left(q\right)$ at jumps

Let $q\in {\mathcal{D}}_y\setminus {\mathcal{D}}_u$ and denote ${\mathcal{G}}_y\left(q\right):=(\xi ,0,e_u,{\eta }_y,{\eta }_u,0,1)$. Then, in view of (21), we have that

$$\begin{array}{ccc}R\left({\mathcal{G}}_y\left(q\right)\right)\hfill & =\hfill & V\left({\xi }^{+}\right)+{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y^{+}\right)|e_y^{+}{|}^2+{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u\right){\left|e_u\right|}^2+{\eta }_y^{+}+{\eta }_u\hfill \\ \hfill & =\hfill & V\left(\xi \right)+{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u\right){\left|e_u\right|}^2+{\eta }_y+{\eta }_u\hfill \\ \hfill & \le \hfill & R\left(q\right)\hfill \end{array}$$

(32)

Similarly, when $q\in {\mathcal{D}}_u\setminus {\mathcal{D}}_y$, we have

$$\begin{array}{ccc}R\left({\mathcal{G}}_u\left(q\right)\right)\hfill & =\hfill & V\left({\xi }^{+}\right)+{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y\right)|e_y{|}^2+{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u^{+}\right){\left|e_u^{+}\right|}^2+{\eta }_y+{\eta }_u^{+}\hfill \\ \hfill & =\hfill & V\left(\xi \right)+{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y\right){\left|e_y\right|}^2+{\eta }_y+{\eta }_u,\hfill \\ \hfill & \le \hfill & R\left(q\right)\hfill \end{array}$$

(33)

where ${\mathcal{G}}_y\left(q\right):=(\xi ,e_y,0,{\eta }_y,{\eta }_u,1,0)$. When $q\in {\mathcal{D}}_y\cap {\mathcal{D}}_u$, $R\left(\mathcal{G}\right(q\left)\right)\le R\left(q\right)$ in view of the last two cases. As a consequence, for all $q\in \mathcal{D}$

$$\begin{array}{c}\hfill R\left(\mathcal{G}\right(q\left)\right)\le R\left(q\right).\end{array}$$

(34)

Dynamics of $R\left(q\right)$ during flows

Let $q\in \mathcal{C}$. We distinguish four cases in the following.

(i)

$q\in \mathcal{C}$ with ${\tau }_y\le T_y$ and ${\tau }_u\le T_u$. Consequently, in view of (26)–(28), we have that

$$\begin{array}{ccc}\hfill \langle \nabla R\left(q\right),F\left(q\right)\rangle & =\hfill & \langle \nabla V\left(\xi \right),{\mathcal{A}}_1\xi +{\mathcal{B}}_1{e}_y+{\mathcal{M}}_1{e}_u\rangle +{\overline{\gamma }}_y{\displaystyle \frac{d{\varphi }_y}{d{\tau }_y}}{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y\right)|e_y|\langle {\displaystyle \frac{\partial }{\partial e_y}}|e_y|,{\mathcal{A}}_2\xi +{\mathcal{M}}_2{e}_u\rangle +{\overline{\gamma }}_u{\displaystyle \frac{d{\varphi }_u}{d{\tau }_u}}{\left|e_u\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u\right)|e_u|\langle {\displaystyle \frac{\partial }{\partial e_u}}|e_u|,{\mathcal{A}}_3\xi +{\mathcal{B}}_3{e}_y\rangle +{\dot{\eta }}_y+{\dot{\eta }}_u\hfill \\ \hfill & ⩽\hfill & -{\epsilon \left|\xi \right|}^2-|{\mathcal{A}}_2{\xi |}^2-|{\mathcal{A}}_3{\xi |}^2-{\delta }_y{\left|y\right|}^2-{\delta }_u{\left|u\right|}^2\hfill \\ \hfill & \hfill & +{\gamma }_y^2|e_y{|}^2+{\gamma }_u^2|e_u{|}^2+{\overline{\gamma }}_y{\left|e_y\right|}^2\left(-(1+{\kappa }_y){\overline{\gamma }}_y({\varphi }_y^2+1)\right)\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right(L_y|e_u|+|{\mathcal{A}}_2\xi \left|\right)\hfill \\ \hfill & \hfill & +{\overline{\gamma }}_u{\left|e_u\right|}^2\left(-(1+{\kappa }_u){\overline{\gamma }}_u({\varphi }_u^2+1)\right)\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right(L_u|e_y|+|{\mathcal{A}}_3\xi \left|\right)+{\Psi }_y(y,e_y,{\eta }_y)\hfill \\ \hfill & \hfill & +{\Psi }_u(u,e_u,{\eta }_u)\hfill \\ \hfill & =\hfill & -{\epsilon \left|\xi \right|}^2-|{\mathcal{A}}_2{\xi |}^2-|{\mathcal{A}}_3{\xi |}^2-{\delta }_y{\left|y\right|}^2-{\delta }_u{\left|u\right|}^2+{\gamma }_y^2{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & +{\gamma }_u^2|e_u{|}^2-(1+{\kappa }_y){\overline{\gamma }}_y^2|e_y{|}^2{\varphi }_y^2-(1+{\kappa }_y){\overline{\gamma }}_y^2{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right|{\mathcal{A}}_2\xi |-(1+{\kappa }_u){\overline{\gamma }}_u^2{\left|e_u\right|}^2{\varphi }_u^2\hfill \\ \hfill & \hfill & -(1+{\kappa }_u){\overline{\gamma }}_u^2|e_u{|}^2+2{\overline{\gamma }}_y{L}_y{\varphi }_y|e_y\left|\right|e_u|\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{L}_u{\varphi }_u|e_y\left|\right|e_u|+2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right|{\mathcal{A}}_3\xi |+{\delta }_y{\left|y\right|}^2-{\alpha }_y{\eta }_y\hfill \\ \hfill & \hfill & +{\delta }_u{\left|u\right|}^2-{\alpha }_u{\eta }_u\hfill \\ \hfill & =\hfill & -{\epsilon \left|\xi \right|}^2-|{\mathcal{A}}_2{\xi |}^2-|{\mathcal{A}}_3{\xi |}^2+{\gamma }_y^2|e_y{|}^2+{\gamma }_u^2{\left|e_u\right|}^2\hfill \\ \hfill & \hfill & -(1+{\kappa }_y){\overline{\gamma }}_y^2|e_y{|}^2{\varphi }_y^2-(1+{\kappa }_y){\overline{\gamma }}_y^2{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{L}_y{\varphi }_y|e_y\left|\right|e_u|+2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right|{\mathcal{A}}_2\xi |\hfill \\ \hfill & \hfill & -(1+{\kappa }_u){\overline{\gamma }}_u^2|e_u{|}^2{\varphi }_u^2-(1+{\kappa }_u){\overline{\gamma }}_u^2{\left|e_u\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{L}_u{\varphi }_u|e_y\left|\right|e_u|+2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right|{\mathcal{A}}_3\xi |-{\alpha }_y{\eta }_y-{\alpha }_u{\eta }_u.\hfill \end{array}$$

(35)

By using the following facts

$$\begin{array}{ccc}\hfill 2{\overline{\gamma }}_y{L}_y{\varphi }_y|e_y\left|\right|e_u|& \le \hfill & {\kappa }_y{\overline{\gamma }}_y^2{\varphi }_y^2|e_y{|}^2+{\displaystyle \frac{L_y^2}{{\kappa }_y^2}}{\left|e_u\right|}^2\hfill \\ \hfill 2{\overline{\gamma }}_u{L}_u{\varphi }_u|e_y\left|\right|e_u|& \le \hfill & {\kappa }_u{\overline{\gamma }}_u^2{\varphi }_u^2|e_u{|}^2+{\displaystyle \frac{L_u^2}{{\kappa }_u^2}}{\left|e_y\right|}^2\hfill \\ \hfill 2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right|{\mathcal{A}}_2\xi |& \le \hfill & {\overline{\gamma }}_y^2{\varphi }_y^2|e_y{|}^2+{\left|{\mathcal{A}}_2\xi \right|}^2\hfill \\ \hfill 2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right|{\mathcal{A}}_3\xi |& \le \hfill & {\overline{\gamma }}_u^2{\varphi }_u^2|e_u{|}^2+{\left|{\mathcal{A}}_3\xi \right|}^2\hfill \end{array}$$

(36)

and the facts that ${\overline{\gamma }}_y^2={\displaystyle \frac{1}{(1+{\kappa }_y)}}({\gamma }_y^2+{\displaystyle \frac{L_u^2}{{\kappa }_u}}+{ϵ}_y)$ and ${\overline{\gamma }}_u^2={\displaystyle \frac{1}{(1+{\kappa }_u)}}({\gamma }_u^2+{\displaystyle \frac{L_y^2}{{\kappa }_y}}+{ϵ}_u)$, we obtain

$$\begin{array}{ccc}\langle \nabla R\left(q\right),F\left(q\right)\rangle \hfill & \le \hfill & -{\epsilon \left|\xi \right|}^2-{ϵ}_y|e_y{|}^2-{ϵ}_u{\left|e_u\right|}^2-{\alpha }_y{\eta }_y\hfill \\ \hfill & \hfill & -{\alpha }_u{\eta }_u.\hfill \end{array}$$

(37)

Then by direct calculations, we can show that

$$\begin{array}{ccc}\langle \nabla R\left(q\right),F\left(q\right)\rangle \hfill & \le \hfill & -\rho R\left(q\right),\hfill \end{array}$$

(38)

where $\rho :=min\{{\displaystyle \frac{\epsilon }{{\lambda }_{max}\left(Q\right)}},{\displaystyle \frac{{ϵ}_y{\lambda }_y}{{\overline{\gamma }}_y}},{\displaystyle \frac{{ϵ}_u{\lambda }_u}{{\overline{\gamma }}_u}},{\alpha }_y,{\alpha }_u\}$ with Q is some arbitrary positive semi definite matrix.

(ii)

$q\in \mathcal{C}$ with ${\tau }_y\ge T_y$ and ${\tau }_u\le T_u$. Then, in view of (26)–(28), we have that

$$\begin{array}{ccc}\hfill \langle \nabla R\left(q\right),F\left(q\right)\rangle & =\hfill & \langle \nabla V\left(\xi \right),{\mathcal{A}}_1\xi +{\mathcal{B}}_1{e}_y+{\mathcal{M}}_1{e}_u\rangle \hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{\varphi }_y\left({\tau }_y\right)|e_y|\langle {\displaystyle \frac{\partial }{\partial e_y}}|e_y|,{\mathcal{A}}_2\xi +{\mathcal{M}}_2{e}_u\rangle \hfill \\ \hfill & \hfill & +{\overline{\gamma }}_u{\displaystyle \frac{d{\varphi }_u}{d{\tau }_u}}|e_u{|}^2+2{\overline{\gamma }}_u{\varphi }_u\left({\tau }_u\right)|e_u|\langle {\displaystyle \frac{\partial }{\partial e_u}}|e_u|,{\mathcal{A}}_3\xi +{\mathcal{B}}_3{e}_y\rangle \hfill \\ \hfill & \hfill & +{\dot{\eta }}_y+{\dot{\eta }}_u\hfill \\ \hfill & ⩽\hfill & -{\epsilon \left|\xi \right|}^2-|{\mathcal{A}}_2{\xi |}^2-|{\mathcal{A}}_3{\xi |}^2-{\delta }_y{\left|y\right|}^2-{\delta }_u{\left|u\right|}^2\hfill \\ \hfill & \hfill & +{\gamma }_y^2|e_y{|}^2+{\gamma }_u^2|e_u{|}^2+2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right|{\mathcal{A}}_2\xi |\hfill \\ \hfill & \hfill & -(1+{\kappa }_u){\overline{\gamma }}_u^2|e_u{|}^2{\varphi }_u^2-(1+{\kappa }_u){\overline{\gamma }}_u^2{\left|e_u\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{L}_y{\varphi }_y|e_y\left|\right|e_u|+2{\overline{\gamma }}_u{L}_u{\varphi }_u|e_y\left|\right|e_u|\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right|{\mathcal{A}}_3\xi |+{\delta }_y{\left|y\right|}^2-{\mu }_y|e_y{|}^2-{\alpha }_y{\eta }_y+{\delta }_u{\left|u\right|}^2\hfill \\ \hfill & \hfill & -{\alpha }_u{\eta }_u\hfill \\ \hfill & =\hfill & -{\epsilon \left|\xi \right|}^2-|{\mathcal{A}}_2{\xi |}^2-|{\mathcal{A}}_3{\xi |}^2+{\gamma }_y^2|e_y{|}^2+{\gamma }_u^2{\left|e_u\right|}^2\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_y{\varphi }_y|e_y\left|\right|{\mathcal{A}}_2\xi |-(1+{\kappa }_u){\overline{\gamma }}_u^2{\left|e_u\right|}^2{\varphi }_u^2\hfill \\ \hfill & \hfill & -(1+{\kappa }_u){\overline{\gamma }}_u^2|e_u{|}^2+2{\overline{\gamma }}_y{L}_y{\varphi }_y|e_y\left|\right|e_u|\hfill \\ \hfill & \hfill & +2{\overline{\gamma }}_u{L}_u{\varphi }_u|e_y\left|\right|e_u|+2{\overline{\gamma }}_u{\varphi }_u|e_u\left|\right|{\mathcal{A}}_3\xi |-{\mu }_y{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & -{\alpha }_y{\eta }_y-{\alpha }_u{\eta }_u.\hfill \end{array}$$

(39)

Then, by using the facts in (36) and in view of the definition of ${\mu }_y$ in (27) and the fact that ${\varphi }_y\left({\tau }_y\right)={\lambda }_y$ for all ${\tau }_y\ge T_y$ since ${\displaystyle \frac{d{\varphi }_y}{d{\tau }_y}}=0$ for all ${\tau }_y\ge T_y$ as shown in (26), we have

$$\begin{array}{ccc}\hfill \langle \nabla R\left(q\right),F\left(q\right)\rangle & \le \hfill & -{\epsilon \left|\xi \right|}^2+{\gamma }_y^2|e_y{|}^2+{\gamma }_u^2|e_u{|}^2+{\overline{\gamma }}_y^2{\varphi }_y^2|e_y{|}^2-(1+{\kappa }_u){\overline{\gamma }}_u^2{\left|e_u\right|}^2{\varphi }_u^2\hfill \\ \hfill & \hfill & -(1+{\kappa }_u){\overline{\gamma }}_u^2|e_u{|}^2+{\kappa }_y{\overline{\gamma }}_y^2{\varphi }_y^2|e_y{|}^2+{\displaystyle \frac{L_y^2}{{\kappa }_y}}|e_u{|}^2+{\kappa }_u{\overline{\gamma }}_u^2{\varphi }_u^2|e_u{|}^2+{\displaystyle \frac{L_u^2}{{\kappa }_u}}{\left|e_y\right|}^2\hfill \\ \hfill & \hfill & +{\overline{\gamma }}_u^2{\varphi }_u^2|e_u{|}^2-\left({\gamma }_y^2+{\displaystyle \frac{L_u^2}{{\kappa }_u}}+(1+{\kappa }_y){\overline{\gamma }}_y^2{\lambda }_y^2+{ϵ}_y\right){\left|e_y\right|}^2-{\alpha }_y{\eta }_y-{\alpha }_u{\eta }_u\hfill \\ \hfill & =\hfill & -{\epsilon \left|\xi \right|}^2-{ϵ}_y|e_y{|}^2-{ϵ}_u{\left|e_u\right|}^2-{\alpha }_y{\eta }_y-{\alpha }_u{\eta }_u\hfill \end{array}$$

(40)

and thus property (38) holds with the same constant $\rho$ as given in (38).

(iii)

$q\in \mathcal{C}$ with ${\tau }_y\le T_y$ and ${\tau }_u\ge T_u$.

By following the same lines as in the previous case, we reach the same property (38).

(iv)

$q\in \mathcal{C}$ with ${\tau }_y\ge T_y$ and ${\tau }_u\ge T_u$. By following the same lines as in the previous case, we reach the same property (38).

As a result, it holds that, for all $q\in C_y\cap C_u$

$$\begin{array}{ccc}\langle \nabla R\left(q\right),F\left(q\right)\rangle \hfill & \le \hfill & -\rho R\left(q\right).\hfill \end{array}$$

(41)

As a result, in view of (34) and (41), the global asymptotic stability of the hybrid system can be deduced. □

8. Illustrative Example

In this section, we apply the approach to a doubly fed induction generator wind turbine in a local power grid system, borrowed from [56]. The grid consists of four components: a wind turbine, a load, an auxiliary generator, and a dispatchable load, as shown in Figure 2 below.

We assume that the load power demand $P^{★}$ is fully provided by the power $P_G$ extracted from the wind turbine. The task of the controller is to adjust the output power according to the change in power demand by regulating the pitch angle of the wind turbine. When the power $P^{★}$ required by the load exceeds the extracted power from the wind turbine, an external power $P_{AG}$ is provided by an auxiliary generator such as a diesel generator. On the other hand, when $P_G>P^{★}$, i.e., if the power generated by the wind turbine is greater than the power demand and the pitch control action is not fast enough, a dispatchable load such as a resistor bank or irrigation system is employed to dissipate the excess power $P_{DL}$. The auxiliary generator and the dispatchable load are orchestrated using an energy management system (EMS) [56].

The linearized model of the considered system is given by [56] (in the absence of external disturbances)

$$\begin{array}{ccc}\hfill \dot{\beta }\left(t\right)& =\hfill & {\displaystyle \frac{-1}{{\tau }_{\beta }}\beta \left(t\right)+\frac{-1}{{\tau }_{\beta }}{\beta }_{cmd}\left(t\right)}\hfill \\ \hfill {\dot{\omega }}_r\left(t\right)& =\hfill & {\displaystyle \frac{M_3}{J}\beta \left(t\right)+(\frac{M_2}{J}+\frac{P^{★}}{J{\omega }_0^2}){\omega }_r\left(t\right),}\hfill \end{array}$$

(42)

where $\beta \left(t\right)$ and ${\omega }_r\left(t\right)$ are the pitch angle and the shaft angular speed of the wind turbine, respectively, and ${\beta }_{cmd}\left(t\right)$ is the command signal from the controller to adjust the pitch angle. The parameters ${\tau }_{\beta }$, J denote the time constant of the wind turbine blade, and the moment of inertia, respectively, $P^{★}$ is the load power demand, and the parameters $J{\omega }_0$ refer to the moment of inertia J at the angular speed linearization ${\omega }_0$. The coefficients $M_1,M_2,M_3$ are functions of the constant parameters of the wind turbine, and their expressions can be found in [56]. We assume that only the angular speed ${\omega }_r$ can be measured. Hence, the state space model of DFIG can be written in the form of (2) with $x\left(t\right)=(\beta \left(t\right),{\omega }_r\left(t\right))$ is the state vector, $u\left(t\right)={\beta }_{cmd}\left(t\right)$ is the control signal and $y\left(t\right)={\omega }_r\left(t\right)$ is the measured output and the matrices $A,B,C$ are given by

$$\begin{array}{ccc}\hfill A& =\hfill & \left[\begin{array}{cc}{\displaystyle \frac{-1}{{\tau }_{\beta }}}& 0\\ {\displaystyle \frac{M_3}{J}}& {\displaystyle \left(\frac{M_2}{J}+\frac{P^{★}}{J{\omega }_0^2}\right)}\end{array}\right],B=\left[\begin{array}{c}{\displaystyle \frac{-1}{{\tau }_{\beta }}}\\ 0\end{array}\right]\hfill \\ \hfill C& =\hfill & \left[\begin{array}{cc}0& 1\end{array}\right].\hfill \end{array}$$

(43)

We stabilize the system by an observer-based controller of the following form

$$\begin{array}{ccc}\hfill \dot{\zeta }\left(t\right)& =\hfill & A\zeta \left(t\right)+Bu\left(t\right)+F\left(y\right(t)-C\zeta (t\left)\right)\hfill \\ \hfill u\left(t\right)& =\hfill & -K\zeta \left(t\right),\hfill \end{array}$$

(44)

where $\zeta \left(t\right)\in {\mathbb{R}}^{n_x}$ denotes the estimated state and $F,K$ are the observer and the controller gain matrices, respectively. Then, by following similar lines as in Section 5, we obtain the hybrid model (18) with

$$\begin{array}{cccc}\hfill {\mathcal{A}}_1& =\hfill & \left[\begin{array}{cc}A& -BK\\ FC& (A-BK-FC)\end{array}\right],\hfill & {\mathcal{B}}_1=\left[\begin{array}{c}0\\ F\end{array}\right]\hfill \\ \hfill {\mathcal{A}}_2& =\hfill & \left[\begin{array}{cc}-CA& CBK\end{array}\right],\hfill & {\mathcal{B}}_2=0.\hfill \end{array}$$

(45)

By following the development in Section 4, we derive the hybrid model (18), where the controller gain K and the observer gain F are given by

$$\begin{array}{ccccc}\hfill K& =\hfill & \left[\begin{array}{cc}2.5821& -1.4120\end{array}\right],F\hfill & =\hfill & {\left[\begin{array}{cc}-10.0249& 13.9974\end{array}\right]}^T.\hfill \end{array}$$

(46)

Then, the LMI condition (22) was found to be feasible with the values $\epsilon =730.9475$, ${\delta }_y=942.2930$, ${\delta }_u=157.7223$, ${\gamma }_y=152.2790$ and ${\gamma }_u=112.5673$. We also found the MATI bounds for the output and input channels are $T_y=0.0103,T_u=0.0140$. We set all the remaining arbitrary parameters at the same value of $0.01$. We ran simulations with the initial condition $\left(x\right(0,0),e(0,0\left)\right)=(1,3,0,0,0,0)$, and we obtained the closed-loop response as shown below. The average inter-transmission times were found to be ${\overline{T}}_y=0.0521$ and ${\overline{T}}_u=0.0417$, while the minimum achieved transmission intervals were $T_{min}^y=0.0103$ and $T_{min}^u=0.0138$. We note that the average inter-transmission times are much larger than the periodic intervals, which demonstrates the effectiveness of the proposed approach.

Figure 3 shows the state trajectories of the plant and the state observer, which all converge to the origin as expected.

The sampling-induced errors of the plant output $e_y\left(t\right)$ and for the control input $e_u\left(t\right)$ are shown in Figure 4, where it can be seen that both $e_y\left(t\right)$ and $e_u\left(t\right)$ are reset to zero at each corresponding sampling instant $t_k^y$ and $t_k^u$, respectively.

The inter-transmission times of the plant output are demonstrated in Figure 5. We note that most of the generated transmission intervals are larger than the enforced dwell-time $T_u$, which reflects the advantage of the proposed methodology.

Similarly, Figure 6 shows the inter-transmission times of the control input $u\left(t\right)$, which are also greater than the guaranteed dwell-time $T_u$.

In order to highlight the advantages of the approach with respect to existing relevant results,

Table 2. Comparison with [42].

Result	Guaranteed Min Sampling Time
[42]	$3.881\times {10}^{-4}$ s
Proposed ETC technique	$0.0135$ s

and

Table 3. Comparison with [43].

Result	Guaranteed Min Sampling Time
[43]	non-zero
Proposed ETC technique	$0.0062$ s

below show the obtained minimum inter-transmission times. It is clear that the proposed approach produces a much larger minimum time between the inter-event times than the approach of [42]. Moreover, the technique of [43] only ensures the existence of a minimum positive time between inter-transmissions, which could be conservative, while the proposed approach provides an explicit expression for such a lower bound.

9. Conclusions

We have investigated the problem of asynchronous event-triggering of the plant output and the control input of LTI systems. The proposed triggering mechanisms are enriched by internal dynamical variables to further reduce the amount of communication over the network. Moreover, strictly positive lower bounds on the triggering instants are imposed to ensure that Zeno phenomena do not occur at either side of the channel. The latter property is established by deriving maximally allowable transmission intervals of time-triggered controllers. The approach is constructed by emulation, where we first stabilize the plant in the absence of a network, and then we take into account the communication constraints. Sufficient conditions have been provided in terms of the feasibility of an LMI condition. The technique has been verified using numerical simulation.

An interesting future extension of this work is to consider more general classes of nonlinear systems, which is a challenging problem and requires different design tools. Moreover, the consideration of other network constraints such as delays and packet dropout is relevant in practice. The robustness against plant disturbances and measurement noises is another important problem to investigate.