Output-Based Dynamic Periodic Event-Triggered Control with Application to the Tunnel Diode System

Abstract

The integration of communication channels with the feedback loop in a networked control system (NCS) is attractive for many applications. A major challenge in the NCS is to reduce transmissions over the network between the sensors, the controller, and the actuators to avoid network congestion. An efficient approach to achieving this goal is the event-triggered implementation where the control actions are only updated when necessary from stability/performance perspectives. In particular, periodic event-triggered control (PETC) has garnered recent attention because of its practical implementation advantages. This paper focuses on the design of stabilizing PETC for linear time-invariant systems. It is assumed that the plant state is partially known; the feedback signal is sent to the controller at discrete-time instants via a digital channel; and an event-triggered controller is synthesized, solely based on the available plant measurement. The constructed event-triggering law is novel and only verified at periodic time instants; it is more adapted to practical implementations. The proposed approach ensures a global asymptotic stability property for the closed-loop system under mild conditions. The overall model is developed as a hybrid dynamical system to truly describe the mixed continuous-time and discrete-time dynamics. The stability is studied using appropriate Lyapunov functions. The efficiency of the technique is illustrated in the dynamic model of the tunnel diode system.

1. Introduction

A networked control system (NCS) is a feedback control system that uses communication networks to carry out data transmission between the sensors, the controller, and the actuators [1]. Due to its numerous benefits, it has been adopted in various domains of applications, such as sensor networks [2], cyber-physical systems [3], multi-agent systems [4], and distributed control systems [5]. However, the reliability of an NCS is still a main challenge compared to a conventional point-to-point connection. The past few decades have seen extensive research on NCSs driven by their inherent challenges. In a conventional digital control system over NCS, the feedback information is sampled, and computed control inputs are then applied to actuators. Traditionally, the sampling instants and execution of control updates are implemented in a periodic time-triggered manner. One drawback of this approach is that periodic controllers can provide unnecessary inputs even when performance is good, which leads to wasteful energy use and actuator wear. Moreover, the design of the periodic sampling period is often conservative and based on the worst-case scenario. Hence, the implantation of such a method in the NCS can result in overloaded and congested traffic over the network. As a result, there will be more frequent packet failures and longer communication delays, which adversely affect the system performance or network reliability. Alternatively, to overcome these limitations, the event-triggered control technique has been proposed in the literature.

Event-triggered control (ETC) is an implementation technique where the transmission instants of the feedback measurements are generated by a state-dependent rule instead of the traditional periodic sampling approach. This allows for more efficient utilization of the limited bandwidth of the shared communication channel. This idea has attracted a great deal of research attention due to its appealing potential in reducing the communication loads among the different components of the system while maintaining satisfactory performance of the closed-loop system. In this context, the ETC has been widely investigated with different control techniques, such as the PID control [6,7,8], state-feedback [9,10], output-based control [11,12,13,14], model-free control [15,16,17], and the consensus of multi-agent systems [18,19,20]. Many developed ETC approaches rely on the continuous verification of the triggering condition to determine the subsequent transmission time; see [21,22,23,24]. However, a major challenge in this type of continuous ETC is to prevent the accumulation of transmission instants, such as the Zeno phenomenon [25]. Alternatively, periodic event-triggered control (PETC) has been proposed, such that the triggering rule is only checked at periodic time instants; it is more adapted to practical implementations and automatically rules out Zeno behavior [13,26,27,28,29,30,31,32].

In this study, we address the issue of output feedback linear time-invariant (LTI) systems under the framework of periodic event-triggered control. It is assumed that only an output of the plant is known and an appropriate periodic ETC is constructed to decide whether to release the transmission at the next periodic instant. The proposed periodic ETC is novel and equipped with a dynamic variable, i.e., dynamic periodic ETC, to further reduce the amount of transmissions. Moreover, the periodic sampling interval is designed based on the approach of [33] to derive the maximally allowable transmission interval (MATI) for the case of time-triggered control. The proposed approach is designed by emulation where we first ignore the effect of the network and stabilize the plant in continuous-time. Then, the sampling due to the network is taken into account, and a periodic ETC mechanism is constructed, such that the closed-loop stability is preserved. The overall system is formulated as a hybrid dynamical system to truly describe the mixed continuous-time and discrete-time dynamics of the system. Sufficient conditions are provided in terms of the linear matrix inequality (LMI) to properly identify the parameters of the event-triggering mechanism in a systematic manner. The stability is investigated by using appropriate Lyapunov functions. The effectiveness of the approach is demonstrated via its application to the tunnel diode circuit system.

The rest of the paper is organized as follows. We present related works in Section 2. Preliminaries are presented in Section 3. The problem is formally stated in Section 4. The hybrid model is presented in Section 5. The main results are presented in Section 6. The application of the technique to the tunnel diode system is presented in Section 7. Conclusions are provided in Section 9.

2. Related Work

The problem of PETC synthesis has been studied in several works; see [29,34,35,36,37,38]. In [34], the PETC problem has been studied for linear systems whose plant dynamics are continuous while the controller dynamics are discrete. In addition, the proposed PETC in [34] considered the effect of quantization as well as sampling in an asynchronous fashion. The PETC technique of [35] has been developed to stabilize linear systems in the presence of sampling and quantization. The idea of this approach is based on constructing abstract models by dividing the state space into finite regions and analyzing the PETC behavior in each region. Similar to [34], the plant is described in continuous-time while the controller is given in discrete-time form. The authors of [36] consider the problem of dynamic PETC for nonlinear multi-agent systems using state feedback control. The approach is developed for both directed and undirected graph topologies under different connection assumptions. Then, the stability is carried out in the framework of hybrid dynamics. In [37], PETC is proposed to stabilize linear systems based on the emulation approach by exploiting model-based computations. The analysis is conducted in two frameworks based on perturbed linear and piecewise linear systems, which lead to different stability guarantees. The authors of [29] construe the PETC mechanism for nonlinear systems, assuming that the controller transmits the control updates to the actuators over multiple decoupled networks. To that end, asynchronous PETC conditions have been developed, such that transmissions over each network are determined by the local triggering rule. Although the considered setup in [29] is quite general, the proposed PETC in [29] is based on the static triggering rule, which may not provide optimal performance in terms of reduced transmissions. Similarly, the PETC approach recently proposed in [38] is designed to stabilize output feedback linear systems based on a static triggering threshold.

We note that the majority of existing works on PETC are adapted to the state feedback control case or based on static triggering rules. The proposed approach in this paper is adapted to the output feedback control case while the PETC relies on internal dynamic variables to allow for a further reduction in transmissions. Moreover, the setup that we consider, the proposed PETC mechanism, the hybrid model, and the obtained stability property are different from previously mentioned works, as explained in Table 1.

Table 1. Novelty of the proposed approach w.r.t. existing results.

	Feedback	Plant	Modeling	Event-Triggering	Stability	Application
	Signals	Dynamics	Framework	Mechanism	Property
[29]	output	nonlinear	hybrid	static	input-to-state	numerical
[34]	output	linear	discrete	static	${\mathcal{L}}_2$-gain	numerical
[35]	output	linear	continuous	static	practical	numerical
[36]	state	nonlinear	hybrid	dynamic	consensus	numerical
[37]	output	linear	discrete	static	${\mathcal{L}}_2$-gain	numerical
[38]	output	linear	continuous	static	${\mathcal{L}}_2$-gain	numerical
Proposed approach	output	linear	hybrid	dynamic	asymptotic	Tunnel diode

Table 1 illustrates the main contributions of the developed ETC approaches compared to the relevant results, consisting of the following main points:

• We construct a novel PETC for linear systems based on available output measurements rather than the full state, as in [36].
• The sampling period is designed as the maximally allowable transmission interval to provide a joint design for both the sampling period and the PETC mechanism, which is not the case in [34,35,38].
• The closed-loop system is modeled as a hybrid dynamical system, which truly describes the existing mixed dynamics, in contrast to [34,35,36,37].
• The PETC mechanism is based on a dynamic internal variable, further saving the communication network compared to the static PETC, as in [29,34,35,37,38].

3. Preliminaries

Let ${\mathbb{R}}_{}^{}:=(-\infty ,\infty )$, ${\mathbb{R}}_{⩾0}^{}:=[0,\infty )$, $\mathbb{N}:=\{0,1,2,\dots \}$ and ${\mathbb{N}}_{>0}:=\{1,2,\dots \}$. A continuous function $\gamma :{\mathbb{R}}_{⩾0}^{}\to {\mathbb{R}}_{⩾0}^{}$ is of class $\mathcal{K}$ if it is zero, at zero, and strictly increasing. It is of class ${\mathcal{K}}_{\infty }$ if, in addition, $\gamma \left(s\right)\to \infty$ as $s\to \infty$. A continuous function $\gamma :{\mathbb{R}}_{⩾0}^2\to {\mathbb{R}}_{⩾0}^{}$ is of class $\mathcal{KL}$ if for each fixed $t\in {\mathbb{R}}_{⩾0}^{}$, $\gamma (·,t)$ is of class $\mathcal{K}$, and $\gamma (s,.)$ is non-increasing and satisfies, for each fixed $s\in {\mathbb{R}}_{⩾0}^{}$, ${\displaystyle \underset{t\to \infty }{lim}\gamma (s,t)=0}$. A function $V:\mathcal{X}\subset {\mathbb{R}}^n\to {\mathbb{R}}_{⩾0}$ is locally Lipschitz-continuous if, for each $x\in \mathcal{X}$, there exists a neighborhood ${\mathcal{U}}_x$ and a constant $M>0$, such that $\left|V\right(y)-V(z\left)\right|⩽M|y-z|$ for all $y,z\in {\mathcal{U}}_x$. Given a locally Lipschitz function $V:\mathcal{X}\subset {\mathbb{R}}^n\to {\mathbb{R}}_{⩾0}$, ${\partial }_x^{C}V\left(x\right)$ denotes the Clarke generalized gradient at $x\in \mathcal{X}$, given by ${\partial }_x^{C}V\left(x\right)=conv\{\zeta \in {\mathbb{R}}^n\mid \exists {\left\{x_i\right\}}_{i\in \mathbb{N}} \mathrm{for} \mathrm{which} {\nabla }_{x}V\left(x_i\right) \mathrm{exists} \mathrm{for} \mathrm{each} i\in \mathbb{N},{lim}_{i\to \infty }{x}_i=x,\mathrm{and}{lim}_{i\to \infty }{\nabla }_{x}V\left(x_i\right)=\zeta \}$. 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. The symbol ★ stands for symmetric blocks. We write $(x,y)\in {\mathbb{R}}_{}^{n_x+n_y}$ to represent the vector ${[x^T,y^T]}^T$ for $x\in {\mathbb{R}}_{}^{n_x}$ and $y\in {\mathbb{R}}_{}^{n_y}$. For a vector $x\in {\mathbb{R}}_{}^{n_x}$, we denote by $\left|x\right|:=\sqrt{x^{T}x}$ its Euclidean norm, and for a matrix $A\in {\mathbb{R}}_{}^{n\times m}$, $\left|A\right|:=\sqrt{{\lambda }_{max}\left(A^{T}A\right)}$. Given a set $\mathcal{A}\subset {\mathbb{R}}_{}^n$ and a vector $x\in {\mathbb{R}}_{}^n$, the distance of x to $\mathcal{A}$ is defined as ${\left|x\right|}_{\mathcal{A}}:={inf}_{y\in \mathcal{A}}|x-y|$.

We consider hybrid systems of the following form [39,40]

$$\begin{array}{c}\hfill \dot{x}=F\left(x\right)x\in \mathcal{C},x^{+}\in G\left(x\right)x\in \mathcal{D},\end{array}$$

(1)

where $x\in {\mathbb{R}}_{}^{n_x}$ is the state, $\mathcal{C}$ is the flow set, F is the flow map, $\mathcal{D}$ is the jump set, and G is the jump map. Solutions to system (1) are defined on hybrid time domains. We call a subset $E\subset {\mathbb{R}}_{\ge 0}^{}\times \mathbb{N}$ a compact hybrid time domain if $E={\bigcup }_{j=0}^{J-1}([t_j,t_{j+1}],j)$ for some finite sequences of times $0=t_0⩽t_1⩽...⩽t_J$ and it is a hybrid time domain if for all $(T,J)\in E,E\cap \left(\right[0,T]\times \{0,1,...,J\left\}\right)$ is a compact hybrid time domain. A hybrid signal is a function defined on a hybrid time domain. For all details of solutions and the definition of maximality, we refer the reader to [39,40].

4. Problem Formulation

We consider plant models with the following dynamics:

$$\begin{array}{ccc}\hfill {\dot{x}}_p& =\hfill & A_p{x}_p+B_{p}u\hfill \\ \hfill y& =\hfill & C_p{x}_p,\hfill \end{array}$$

(2)

where $x_p\in {\mathbb{R}}_{}^{n_p}$ is the plant state, $u\in {\mathbb{R}}_{}^{n_u}$ is the control input, $y\in {\mathbb{R}}_{}^{n_y}$ is the measured output, and $A_p,B_p,C_p,E_p$ are matrices of appropriate dimensions. The plant is stabilized by the following dynamic controller:

$$\begin{array}{ccc}\hfill {\dot{x}}_c& =\hfill & A_c{x}_c+B_c\widehat{y}\hfill \\ \hfill u& =\hfill & C_c{x}_c+D_c\widehat{y}\hfill \end{array}$$

(3)

where $x_c\in {\mathbb{R}}_{}^{n_c}$ is the controller state, $\widehat{y}\in {\mathbb{R}}_{}^{n_y}$ is the last transmitted value of y, and $A_c,B_c,C_c,D_c$ are matrices of appropriate dimensions. The feedback law (3) is designed by emulation; that is, we first stabilize the plant (2) in continuous time, assuming perfect communication, i.e., $\widehat{y}=y$. Then, we take into account the sampling effects.

The implementation scenario is shown in Figure 1. We consider the case where the controller is co-located with the plant while the sensors and the controller communicate over a shared network. We assume that the plant state $x_p$ is not available for measurements and only an output $y\left(t\right)$ can be transmitted to the controller. In particular, we consider that the output $y\left(t\right)$ is sampled at periodic sampling times $t_i^s,i\in \mathbb{N}$. Then, an event-triggering mechanism is employed to decide whether to submit the output value $y\left(t_i^s\right)$, where the time instants at which $y\left(t_i^s\right)$ is released are denoted by $t_j^y,j\in \mathbb{N}$, leading to the so-called periodic event-triggered control (PETC), and we refer by $\widehat{y}\left(t_j^y\right)$ the most recent value of $y\left(t_i^s\right)$ at the controller at time $t_j^y,j\in \mathbb{N}$; see Figure 1. Hence, if we define

$$\begin{array}{ccc}\hfill {\mathcal{T}}_s& =\hfill & \left\{t_i^s\right\},i\in \mathbb{N}\hfill \\ \hfill {\mathcal{T}}_y& =\hfill & \left\{t_j^y\right\},j\in \mathbb{N},\hfill \end{array}$$

(4)

where ${\mathcal{T}}_s$ and ${\mathcal{T}}_y$ denote the increasing sequences of periodic time instants and transmission instants, respectively. Then, it holds that ${\mathcal{T}}_y\subseteq {\mathcal{T}}_s$.

Figure 1. Periodic event-triggered output feedback control; (solid line) continuous-time; (dotted line) periodic instants; (dash line) event-triggered instants.

It is important to mention that the periodic event-triggering mechanism is assumed to have access to both the actual output value, i.e., $y\left(t\right)$, and the last transmitted value, $\widehat{y}\left(t_j^k\right),j\in \mathbb{N}$.

The objectives of this paper include

• The synthesis of the periodic sampling interval and periodic event-triggered controller by emulation;
• Derivation of the hybrid dynamical model of the overall system;
• Sufficient conditions to ensure the closed-loop stability;
• Preventing the occurrence of Zeno behavior.

5. Hybrid Model

In this section, we derive the dynamic behavior of the closed-loop system and formulate it as a hybrid dynamical system. We define the sampling error $e_s\left(t\right):{\mathbb{R}}_{}^{n_y}\to {\mathbb{R}}_{}^{n_y}$ and the network-induced error $e_y\left(t\right):{\mathbb{R}}_{}^{n_y}\to {\mathbb{R}}_{}^{n_y}$ between two transmission times as follows, for all $t\in [t_j^y,t_{j+1}^y)$

$$\begin{array}{cccc}\hfill e_s\left(t\right)& :=\hfill & y\left(t_i^s\right)-y\left(t\right)\hfill & \forall t\in [t_i^s,t_{i+1}^s),i\in \mathbb{N}\hfill \\ \hfill e_y\left(t\right)& :=\hfill & \widehat{y}\left(t_j^y\right)-y\left(t_i^s\right)\hfill & \forall t\in [t_j^y,t_{j+1}^y),i,j\in \mathbb{N}.\hfill \end{array}$$

(5)

Between two periodic sampling times $[t_i^s,t_{i+1}^s]$, the sampled output $y\left(t_i^s\right)$ is kept constant using ZOH. At each periodic sampling time, $t_i^s,i\in \mathbb{N}$, the value of $y\left(t_i^s\right)$ is reset to $y\left(t\right)$. Moreover, between two transmission instants, $[t_j^y,t_{j+1}^y]$, the last transmitted value of the output $y\left(t_j^y\right)$ is kept constant using ZOH, and at each transmission instant, $t_j^y,j\in \mathbb{N}$, $e_y\left(t\right)$ is reset to $y\left(t_i^s\right)$.

We define the total error $e\left(t\right)$ as the difference between the last transmitted value of the output ${\widehat{y}}_q\left(t_j^y\right)$ and the current output measurement $y\left(t\right)$:

$$\begin{array}{ccc}\hfill e\left(t\right)& :=\hfill & \widehat{y}\left(t_j^y\right)-y\left(t\right)\forall t\in [t_j^y,t_{j+1}^y)\hfill \\ \hfill & =\hfill & e_s\left(t\right)+e_y\left(t\right).\hfill \end{array}$$

(6)

Then, it holds that

$$\begin{array}{cccc}\hfill \dot{e}\left(t\right)& =\hfill & -\dot{y}=-C_p{\dot{x}}_p\hfill & t\in [t_j^y,t_{j+1}^y)\hfill \\ \hfill e\left(t_j^{y^{+}}\right)& =\hfill & e_s\left(t_j^{y^{+}}\right)+e_y\left(t_j^{y^{+}}\right)\hfill \\ \hfill & =\hfill & 0.\hfill \end{array}$$

(7)

The last property implies that the total error $e\left(t\right)$ is reset to zero at each transmission instant $t_j^y,j\in \mathbb{N}$ since $y\left(t_i^s\right)$ is updated to $y\left(t\right)$ at each $t_j^y,j\in \mathbb{N}$.

Let $x=(x_p,x_c)\in {\mathbb{R}}_{}^{n_x}$. Then, in view of (2), (3), (6), we obtain

$$\begin{array}{ccc}\hfill \dot{x}& =\hfill & \left[\begin{array}{cc}{A}_p+B_p{D}_c{C}_p& B_p{C}_c\\ B_c{C}_p& A_c\end{array}\right]x+\left[\begin{array}{c}{B}_p{D}_c\\ B_c\end{array}\right]e\hfill \\ \hfill & =:\hfill & {\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill \dot{e}& =\hfill & \left[\begin{array}{cc}-C_p(A_p+B_p{D}_c{C}_p)& -C_p{B}_p{C}_c\end{array}\right]x+\left[\begin{array}{c}-C_p{B}_p{D}_c\end{array}\right]e\hfill \\ \hfill & =:\hfill & {\mathcal{A}}_{2}x+{\mathcal{B}}_{2}e.\hfill \end{array}$$

(9)

We define two auxiliary time variables ${\tau }_s,{\tau }_y:{\mathbb{R}}_{\ge 0}^{}\to {\mathbb{R}}_{\ge 0}^{}$, as follows:

$$\begin{array}{cccc}\hfill {\dot{\tau }}_s\left(t\right)& =\hfill & 1\hfill & t\in [t_i^s,t_{i+1}^s)\hfill \\ \hfill {\tau }_s\left(t_i^{s^{+}}\right)& =\hfill & 0\hfill & t\in {\left\{t_i^s\right\}}_{i\in \mathbb{N}}\hfill \end{array}$$

(10)

and

$$\begin{array}{cccc}\hfill {\dot{\tau }}_y\left(t\right)& =\hfill & 1\hfill & t\in [t_j^y,t_{j+1}^y)\hfill \\ \hfill {\tau }_y\left(t_j^{y^{+}}\right)& =\hfill & 0\hfill & t\in {\left\{t_j^y\right\}}_{j\in \mathbb{N}·}\hfill \end{array}$$

(11)

The time variable ${\tau }_s$ will be used to describe the time between two periodic sampling instants, $[t_i^s,t_{i+1}^s]$, and it is reset to zero at each periodic instance, $t_i^s,i\in \mathbb{N}$. Similarly, the time variable ${\tau }_y$ will be used to track the time between two transmission instants $[t_j^y,t_{j+1}^y]$ and it is reset to zero at each transmission instance $t_j^y,j\in \mathbb{N}$. These two time variables, ${\tau }_s$ and ${\tau }_y$, will be helpful in constructing the hybrid dynamical model of the system, as explained in the sequel.

In order to complete the description of the overall system, we outline below the general structure of the proposed periodic event-triggering mechanism, which will be clearly developed in the next section. We synthesize a PETC based on a dynamic variable $\eta$, which has the following dynamics; see [41,42,43],

$$\begin{array}{cccc}\hfill \dot{\eta }\left(t\right)& =\hfill & \mathsf{\Psi }(y,e,\eta )\hfill & t\in [t_i^s,t_{i+1}^s)\hfill \\ \hfill \eta \left(t^{+}\right)& =\hfill & g_s(y,e,\eta )\hfill & t\in {\mathcal{T}}_s\setminus {\mathcal{T}}_y\hfill \\ \hfill \eta \left(t^{+}\right)& =\hfill & g_y(y,e,\eta )\hfill & t\in {\mathcal{T}}_y,\hfill \end{array}$$

(12)

where functions $\mathsf{\Psi },g_s$, and $g_y$ will be specified in Section 6. Note that functions $\mathsf{\Psi },{\eta }_s$, and ${\eta }_y$ only depend on the locally available information $(y,e,\eta )$ at the event-triggering mechanism. The transmission instants are generated by the following mechanism:

$$t_{j+1}^y=min\{t>t_j^y|t\in {\mathcal{T}}_s\wedge g_s(y,e,\eta )⩽0\},$$

(13)

where $t_0^y=0$.

In view of (7)–(12), we obtain the following impulsive model:

$$\begin{array}{c}\left.\begin{array}{ccc}\hfill \dot{x}& =& {\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e\hfill \\ \hfill \dot{e}& =& {\mathcal{A}}_{2}x+{\mathcal{B}}_{2}e\hfill \\ \hfill \dot{\eta }\left(t\right)& =& \mathsf{\Psi }(y,e,\eta )\hfill \\ \hfill {\dot{\tau }}_s\left(t\right)& =& 1\hfill \\ \hfill {\dot{\tau }}_y\left(t\right)& =& 1\hfill \end{array}\right\}t\notin {\mathcal{T}}_s\hfill \\ \begin{array}{c}u=C_c{x}_c+D_c{\widehat{y}}_q\hfill \\ y=C_p{x}_p\hfill \end{array}\hfill \\ \begin{array}{cc}\left.\begin{array}{ccc}\hfill {\tau }_s\left(t^{+}\right)& =& 0\hfill \\ \hfill \eta \left(t^{+}\right)& =& g_s(y,e,\eta )\hfill \end{array}\right\}\hfill & t\in {\mathcal{T}}_s\setminus {\mathcal{T}}_y\hfill \\ \left.\begin{array}{ccc}\hfill e\left(t^{+}\right)& =& e_q\left(t\right)\hfill \\ \hfill \eta \left(t^{+}\right)& =& g_y(y,e,\eta )\hfill \\ \hfill {\tau }_s\left(t^{+}\right)& =& 0\hfill \\ \hfill {\tau }_y\left(t^{+}\right)& =& 0\hfill \end{array}\right\}\hfill & t\in {\mathcal{T}}_y\hfill \end{array}\hfill \end{array}$$

(14)

Let $\xi :=(x,e,\eta ,{\tau }_s,{\tau }_y)\in \mathbb{X}$ be the concatenation of the state variables, with $\mathbb{X}={\mathbb{R}}_{}^{n_x}\times {\mathbb{R}}_{}^{n_y}\times {\mathbb{R}}_{⩾0}^{}\times {\mathbb{R}}_{⩾0}^{}\times {\mathbb{R}}_{⩾0}^{}$. Then, we obtain the hybrid dynamical system:

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

(15)

where the flow set ${\mathcal{C}}_s$ and the jump ${\mathcal{D}}_s$ are defined as

$$\begin{array}{cccc}\hfill {\mathcal{C}}_s& :=\hfill & \{\xi \in \mathbb{X}:\hfill & {\tau }_s\in [0,T]\}\hfill \\ \hfill {\mathcal{D}}_s& :=\hfill & \{\xi \in \mathbb{X}:\hfill & {\tau }_s=T\},\hfill \end{array}$$

(16)

where $T>0$ is the periodic sampling interval to be designed. We also define the jump set ${\mathcal{D}}_y\subset {\mathcal{D}}_s$ to identify the transmission instants as follows:

$$\begin{array}{cccc}\hfill {\mathcal{D}}_y& :=\hfill & \{\xi \in \mathbb{X}:\hfill & {\tau }_s=T\mathrm{and}{g}_s\left(t\right)⩽0\}.\hfill \end{array}$$

(17)

It is evident from (16) and (17) that ${\mathcal{D}}_y\subset {\mathcal{D}}_s$. The flow map $\mathcal{F}\left(\xi \right)$ and the jump map $\mathcal{G}\left(\xi \right)$ in (15) are given by

$$\begin{array}{ccc}\hfill \mathcal{F}\left(\xi \right)& =\hfill & \left(\begin{array}{c}{\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e\\ {\mathcal{A}}_{2}x+{\mathcal{B}}_{2}e\\ \mathsf{\Psi }(y,e,\eta )\\ 1\\ 1\end{array}\right)\hfill \end{array}$$

(18)

and

$$\mathcal{G}\left(\xi \right):=\left\{\begin{array}{cc}\left\{G_s\left(\xi \right)\right\}\hfill & \xi \in {\mathcal{D}}_s\hfill \\ \left\{G_y\left(\xi \right)\right\}\hfill & \xi \in {\mathcal{D}}_y\hfill \\ \left\{G_s\left(\xi \right),G_y\left(\xi \right)\right\}\hfill & \xi \in {\mathcal{D}}_s\cap {\mathcal{D}}_y\hfill \\ \varnothing \hfill & \xi \notin {\mathcal{D}}_s\cup {\mathcal{D}}_{\mu }\hfill \end{array}\right.$$

(19)

with

$$\begin{array}{cccccc}\hfill G_s\left(\xi \right)& :=\hfill & \left(\begin{array}{c}x\\ e\\ g_s\\ 0\\ {\tau }_y\end{array}\right),\hfill & G_y\left(\xi \right)\hfill & :=\hfill & \left(\begin{array}{c}x\\ 0\\ g_y\\ 0\\ 0\end{array}\right)\hfill \end{array}$$

(20)

The system flows on ${\mathcal{C}}_s$ when ${\tau }_s⩽T$, i.e., between two periodic instants; otherwise, the system experiences a jump. The jump map in (20) can be interpreted as follows. When $\xi \in {\mathcal{D}}_s\setminus {\mathcal{D}}_y$, only variables ${\tau }_s$ and $\eta$ are updated but no transmission is generated. When $\xi \in {\mathcal{D}}_y$, implying that the triggering condition is violated, variables $e,\eta ,{\tau }_s,{\tau }_y$ are updated.

Remark 1. 

The hybrid dynamical model is built on the following boundary conditions:

•    The plant model (2) is linear time-invariant.
•    The plant model is controllable and observable.
•    The communication network is noiseless and delay-free.
•    The sampling-induced error $e\left(t\right)$ is reset to zero at each triggering instant $t_k$.
•    The control input $u\left(t_k\right)$ is kept constant between two transmissions $t\in [t_k,t_{k+1}]$ by ZOH.
•    The plant state $x_p\left(t\right)$ and the observer state $x_c\left(t\right)$ do not change at transmissions.

6. Main Results

We present here the main results. First, we state the following lemma on system (15).

Lemma 1. 

Consider system (15). If there exist ${\epsilon }_x,{\epsilon }_y,\gamma >0$, and a positive definite symmetric real matrix P, such that

$$\left[\begin{array}{cc}{\mathcal{A}}_1^{T}P+P{\mathcal{A}}_1+{\epsilon }_x{\mathbb{I}}_{n_x}+{\mathcal{A}}_2^T{\mathcal{A}}_2+{\epsilon }_y{\tilde{C}}_p^T{\tilde{C}}_p& P{\mathcal{B}}_1\\ {\mathcal{B}}_1^{T}P& -{\gamma }^2{\mathbb{I}}_{n_e}\end{array}\right]⩽0,$$

(21)

where ${\tilde{C}}_p:=\left[C_{p}0\right]$, then the Lyapunov function candidate $V\left(x\right)=x^{T}Px$ satisfies, for all $e\in {\mathbb{R}}_{}^{n_e}$ and almost all $x\in {\mathbb{R}}_{}^{n_x}$

$$\begin{array}{ccc}\hfill ⟨\nabla V\left(x\right),{\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e⟩& \le \hfill & -{\epsilon }_x{\left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-{\epsilon }_y{\left|y\right|}^2+{\gamma }^2{\left|e\right|}^2.\hfill \end{array}$$

(22)

Proof of Lemma 1 

Let $V\left(x\right)=x^{T}Px$. Consequently, it holds that, for all $e\in {\mathbb{R}}_{}^{n_e}$ and almost all $x\in {\mathbb{R}}_{}^{n_x}$,

$$\begin{array}{ccc}\hfill ⟨\nabla V\left(x\right),{\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e⟩& =\hfill & x^T({\mathcal{A}}_1^{T}P+P{\mathcal{A}}_1)x\hfill \\ \hfill & \hfill & +x^{T}P{\mathcal{B}}_{1}e+e^T{\mathcal{B}}_1^{T}Px.\hfill \end{array}$$

(23)

By post- and pre-multiplying LMIs (21), respectively, by the state vector $(x,e)$, and its transpose, we obtain

$$\begin{array}{c}\hfill x^T({\mathcal{A}}_1^{T}P+P{\mathcal{A}}_1)x+x^{T}P{\mathcal{B}}_{1}e+e^T{\mathcal{B}}_1^{T}Px\le -{\epsilon }_x{x}^{T}x\\ \hfill -x^T{\mathcal{A}}_2^T{\mathcal{A}}_{2}x-{\epsilon }_y{x}^T{\tilde{C}}_p^T{\tilde{C}}_{p}x+{\gamma }^2{e}^{T}e\end{array}$$

(24)

which implies

$$\begin{array}{c}\hfill x^T({\mathcal{A}}_1^{T}P+P{\mathcal{A}}_1)x+x^{T}P{\mathcal{B}}_{1}e+e^T{\mathcal{B}}_1^{T}Px\le \\ \hfill -{\epsilon }_x{\left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-{\epsilon }_y|{\tilde{C}}_p{x|}^2+{\gamma }^2{\left|e\right|}^2\\ \hfill =-{\epsilon }_x{\left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-{\epsilon }_y{\left|y\right|}^2+{\gamma }^2{\left|e\right|}^2\end{array}$$

(25)

and the conclusion of Lemma 1 holds. □

Lemma 1 establishes an ${\mathcal{L}}_2$-gain stability property for the system $\dot{x}={\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e$ from $\left|e\right|$ to $\left(\right|{\mathcal{A}}_{2}x|,|y\left|\right)$; see [9,33,42].

6.1. Event-Triggering Mechanism

We define $W\left(e\right):=\left|e\right|$, then in view of (15), it holds that for all $x\in {\mathbb{R}}_{}^{n_x}$ and almost all $e\in {\mathbb{R}}_{}^{n_e}$

$$\begin{array}{c}\hfill ⟨\nabla W\left(e\right),{\mathcal{A}}_{2}x+{\mathcal{B}}_{2}e⟩⩽|{\mathcal{A}}_{2}x|+L|e|,\end{array}$$

(26)

where $L:=|{\mathcal{B}}_2|$.

The dynamics of the triggering function $\eta$ in (12) are defined by functions $\mathsf{\Psi }$, ${\eta }_s$, and ${\eta }_y$, which are given by

$$\begin{array}{ccc}\hfill \mathsf{\Psi }(y,e,\eta )& :=\hfill & {\epsilon }_y{\left|y\right|}^2-\vartheta \eta \hfill \\ \hfill g_s(y,e,\eta )& :=\hfill & \gamma (\lambda -\frac{1}{\lambda }){\left|e\right|}^2+\eta \hfill \\ \hfill g_y(y,e,\eta )& :=\hfill & {\gamma \lambda \left|e\right|}^2+\eta \hfill \end{array}$$

(27)

where $\lambda \in (0,1)$, $\tilde{\lambda }\in [\lambda ,{\lambda }^{-1})$, $\tilde{\gamma }:={\gamma }^2+{\gamma }^2{\tilde{\lambda }}^2+2\gamma \tilde{\lambda }\tilde{L}$ with $\tilde{L}:=L+\nu$ for any $\nu >0$ and $L=|{\mathcal{B}}_2|$, and the constant $\gamma$ comes from Lemma 1. The sampling period T is designed as the maximally allowable transmission interval (MATI) of time-triggered systems [33], which leads to

$$T(\lambda ,\tilde{\lambda },\gamma ,\tilde{L}):=\left\{\begin{array}{cc}\frac{1}{\tilde{L}r}arctan\left(\frac{r(1-\lambda \tilde{\lambda })}{\frac{\gamma }{\tilde{L}}(\lambda +\tilde{\lambda })+1+\lambda \tilde{\lambda }}\right)\hfill & \gamma >\tilde{L}\hfill \\ \frac{1}{\tilde{L}}\frac{1-\lambda \tilde{\lambda }}{\lambda \tilde{\lambda }+\lambda +\tilde{\lambda }+1}\hfill & \gamma =\tilde{L}\hfill \\ \frac{1}{\tilde{L}r}\arctan \left(\frac{r(1-\lambda \tilde{\lambda })}{\frac{\gamma }{\tilde{L}}(\lambda +\tilde{\lambda })+1+\lambda \tilde{\lambda }}\right)\hfill & \gamma <\tilde{L}\hfill \end{array}\right.$$

(28)

with $r:=\sqrt{\left|{\left(\frac{\gamma }{\tilde{L}}\right)}^2-1\right|}$. Note that when $\tilde{\lambda }=\lambda$ in (28), we recover the MATI bound of time-triggered controllers in [33]. By designing the sampling period T as the MATI bound, we opt to further reduce the amount of transmissions by using the PETC mechanism.

Remark 2. 

It is important to note that in view of (27), we have that $\dot{\eta }\left(t\right)={\epsilon }_y{\left|y\right|}^2-\vartheta \eta ⩾-\vartheta \eta$. Moreover, $\eta \left(t\right)$ is reset to $g_s(y,e,\eta )$ when $g_s(y,e,\eta )>0$ and $\eta \left(t\right)$ are reset to $g_y(y,e,\eta )$, which is strictly positive when $g_s(y,e,\eta )⩽0$. Consequently, by using the comparison principle, it holds that $\eta \left(t\right)⩾0$ for all $t\in {\mathbb{R}}_{⩾0}^{}$. This property is crucial for establishing the stability of the closed-loop system, as will be shown later.

6.2. Stability Results

We obtain the following results.

Theorem 1. 

Consider system (15) with the flow and jump sets, as in (16). Suppose that the LMI (21) in Lemma 1 is satisfied. Then, there exists a $\mathcal{KL}$ function β, such that any solution $\xi (t,j)\in \mathbb{X}$ satisfies

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

(29)

Proof of Theorem 1. 

Let $\varphi$ be the solution to the differential system

$$\begin{array}{ccc}\frac{d\varphi }{d\tau }\hfill & :=\hfill & \left\{\begin{array}{cc}-2\tilde{L}\varphi \left(\tau \right)-\gamma ({\varphi }^2\left(\tau \right)+1)\hfill & \tau \in [0,T]\hfill \\ 0\hfill & \tau >T,\hfill \end{array}\right.\hfill \end{array}$$

(30)

where $\tilde{L}:=L+\nu$ with $L=|{\mathcal{B}}_2|$ and for any $\nu >0$, and the constant $\gamma$ comes from Lemma 1. The time constant T is the time it takes for $\varphi$ to decrease from $\varphi \left(0\right)=\frac{1}{\lambda }$ to $\varphi \left(T\right)=\tilde{\lambda }$ with $\lambda \in (0,1)$ and $\tilde{\lambda }\in [\lambda ,{\lambda }^{-1})$. This time is given by (28), which can be derived by following similar lines, as in [33].

We define for all $\xi \in {\mathcal{C}}_s\cup {\mathcal{D}}_s$

$$R\left(\xi \right):=V\left(x\right)+{\gamma \varphi \left(\tau \right)\left|e\right|}^2+\eta ,$$

(31)

where $V\left(x\right):=x^{T}Px$ for all $(x,e)\in {\mathbb{R}}_{}^{n_x+n_e}$ with P comes from Lemma 1. Note that, in view of (30), $\varphi \left(\tau \right)>0$ for all $(t,j)\in \mathrm{dom}\xi$. Hence, since $\eta ⩾0$ for all $(t,j)\in \mathrm{dom}\xi$, as explained in Remark 2, we deduce that R is positive-definite and radially unbounded and, thus, R can be used as a Lyapunov function candidate to investigate the behavior of the hybrid system (15). We study the dynamics of R during jumps and flows.

Dynamics of  R  during flows

Let $\xi \in {\mathcal{C}}_s$. Consequently, in view of (12), (25), (26), (30), (31), we obtain (recall that $\tilde{L}=L+\nu$)

$$\begin{array}{c}⟨\nabla R,F(\xi ,w)⟩=⟨\nabla V\left(x\right),{\mathcal{A}}_{1}x+{\mathcal{B}}_{1}e⟩\hfill \\ +\gamma \frac{d\varphi }{d\tau }{\left|e\right|}^2+2\gamma \varphi \left(\tau \right)\left|e\right|⟨\frac{\partial }{\partial e}\left|e\right|,g(x,e)⟩+\dot{\eta }\hfill \\ ⩽-{\epsilon }_x{\left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-{\epsilon }_y{\left|y\right|}^2+{\gamma }^2{\left|e\right|}^2\hfill \\ +{\gamma \left|e\right|}^2\left(-2\tilde{L}\varphi \left(\tau \right)-\gamma ({\varphi }^2\left(\tau \right)+1)\right)\hfill \\ +2\gamma \varphi \left(\tau \right)\left|e\right|\left(L\right|e|+|{\mathcal{A}}_{2}x)+\mathsf{\Psi }(y,e,\eta ,\tau )\hfill \\ =-{\epsilon }_x{\left|x\right|}^2-|{\mathcal{A}}_2{x|}^2-{\epsilon }_y{\left|y\right|}^2-2\nu \gamma \varphi \left(\tau \right){\left|e\right|}^2\hfill \\ +2\gamma \varphi \left(\tau \right)\left|e\right||{\mathcal{A}}_{2}x|-{\gamma }^2{\varphi }^2\left(\tau \right){\left|e\right|}^2+\mathsf{\Psi }(y,e,\eta ,\tau ).\hfill \end{array}$$

By using the fact that $2\gamma \varphi \left(\tau \right)\left|e\right||{\mathcal{A}}_{2}x|⩽{\gamma }^2{\varphi }^2{\left(\tau \right)\left|e\right|}^2+{\left|{\mathcal{A}}_{2}x\right|}^2$ and since $\mathsf{\Psi }(y,e,\eta ,\tau )={\epsilon }_y{\left|y\right|}^2-\vartheta \eta$ in view of (27), we have that for all $\xi \in \mathcal{C}$ and $\tau \in [0,T]$

$$\begin{array}{c}\hfill ⟨\nabla R,F(\xi ,w)⟩⩽-{\epsilon }_x{\left|x\right|}^2-2\nu \gamma \varphi \left(\tau \right){\left|e\right|}^2-\vartheta \eta .\end{array}$$

(32)

By using the fact that ${\lambda }_{min}{\left(P\right)\left|x\right|}^2⩽V\left(x\right)⩽{\lambda }_{max}\left(P\right){\left|x\right|}^2$, it holds that, for all $\xi \in {\mathcal{C}}_s$

$$\begin{array}{ccc}⟨\nabla R,F(\xi ,w)⟩\hfill & ⩽\hfill & -\frac{{\epsilon }_{x}V\left(x\right)}{{\lambda }_{max}\left(P\right)}-2\nu \gamma \varphi \left(\tau \right){\left|e\right|}^2-\vartheta \eta \hfill \\ \hfill & \stackrel{\left(31\right)}{⩽}\hfill & -\rho R\left(\xi \right),\hfill \end{array}$$

(33)

where $\rho :=min\{\frac{(1-ϵ){\epsilon }_x}{{\lambda }_{max}\left(P\right)},2\nu ,\vartheta \}$.

Dynamics of  R  at jumps

In view of the jump map in (20), we distinguish three different situations.

• Let $\xi \in {\mathcal{D}}_s\setminus {\mathcal{D}}_y$, which implies that $g_s(y,e,\eta )⩾0$, and no transmission occurs at this periodic instance, $t_i^s$, and $\eta$ is updated to ${\eta }_s$. We have that, in view of (15) and (20), and by using the fact that $\varphi \left(0\right)={\lambda }^{-1}$:

  $$\begin{array}{ccc}R\left(G\right(\xi \left)\right)\hfill & =& V\left(x^{+}\right)+\gamma \varphi \left({\tau }^{+}\right){\left|e^{+}\right|}^2+{\eta }^{+}\hfill \\ \hfill & =& V\left(x\right)+{\gamma \varphi \left(0\right)\left|e\right|}^2+g_s(y,e,\eta )\hfill \\ \hfill & =& V\left(x\right)+\frac{\gamma }{\lambda }{\left|e\right|}^2+g_s(y,e,\eta )\hfill \\ \hfill & =& V\left(x\right)+\frac{\gamma }{\lambda }{\left|e\right|}^2+{\left(\gamma \lambda \right|e|}^2+\eta -\frac{\gamma }{\lambda }{\left|e\right|}^2)\hfill \\ \hfill & =& V\left(x\right)+{\gamma \lambda \left|e\right|}^2+\eta \hfill \\ \hfill & ⩽& V\left(x\right)+{\gamma \varphi \left(\tau \right)\left|e\right|}^2+\eta \hfill \\ \hfill & ⩽& R\left(\xi \right).\hfill \end{array}$$

  (34)
• Let $\xi \in {\mathcal{D}}_y$, which implies that $g_s(y,e,\eta )⩽0$, a transmission occurs at this periodic instance $t_i^s$, and $\eta$ is updated to $g_y$. We have that, in view of (15) and (20), and by using the fact that $\varphi \left(0\right)={\lambda }^{-1}$:

  $$\begin{array}{ccc}R\left(G\right(\xi \left)\right)\hfill & =& V\left(x^{+}\right)+\gamma \varphi \left({\tau }^{+}\right){\left|e^{+}\right|}^2+{\eta }^{+}\hfill \\ \hfill & =& V\left(x\right)+g_y(y,e,\eta )\hfill \\ \hfill & =& V\left(x\right)+{\gamma \lambda \left|e\right|}^2+\eta \hfill \\ \hfill & ⩽& V\left(x\right)+{\gamma \varphi \left(\tau \right)\left|e\right|}^2+\eta \hfill \\ \hfill & ⩽& R\left(\xi \right).\hfill \end{array}$$

  (35)
• Let $\xi \in {\mathcal{D}}_s\cap {\mathcal{D}}_y$, i.e., a transmission is synchronized with a periodic sampling instant $t_i^s$. Based on (34) and (35), it is straightforward to see that $R\left(G\right(\xi \left)\right)⩽R\left(\xi \right)$ in these cases.

As a result, we deduce that, for all $\xi \in \mathcal{D}$, the relation $R\left(G\right(\xi \left)\right)⩽R\left(\xi \right)$ is valid. Then, the property $R\left(G\right(\xi \left)\right)⩽R\left(\xi \right)$ holds for all $\xi \in {\mathcal{D}}_s$.

Consequently, in view of (33) and (35), by using similar arguments as in the end of the proof of Theorem 1 in [33], we deduce that the closed-loop system is globally asymptotically stable, and the conclusion of Theorem 1 holds. □

Theorem 1 implies that the closed-loop system (15) is globally asymptotically stable under the proposed PETC.

Remark 3. 

It is clear from the proposed PETM (13) that there exists a trade-off between the periodic sampling interval T and the generated amount of transmissions. That is, when the value of T is enlarged, the generated number of transmissions will be increased and vice versa. This trade-off can be adjusted by the user to satisfy the desirable performance of the PETC.

7. Event-Triggered Control of the Tunnel Diode System

To illustrate the benefit of the approach, the technique is applied to the tunnel’s diode circuit system; see Figure 2.

Figure 2. Schematic of the tunnel diode circuit.

The dynamic model of this system is given by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}{V}_{C_1}\left(t\right)}& =\hfill & -\frac{1}{R_L{C}_1}\left(V_{C_1}\left(t\right)-V_{C_2}\left(t\right)\right)+\frac{1}{C_1}{i}_E\left(t\right)\hfill \\ \hfill {\displaystyle \frac{d}{dt}{V}_{C_2}\left(t\right)}& =\hfill & \frac{1}{R_L{C}_2}\left(V_{C_1}\left(t\right)-V_{C_2}\left(t\right)\right)-\frac{a}{C_2}{V}_{C_2}\left(t\right)\hfill \\ \hfill {\displaystyle \frac{d}{dt}{i}_E\left(t\right)}& =\hfill & \frac{1}{E}\left(-V_{C_1}\left(t\right)-R_E{i}_E\left(t\right)+u\left(t\right)\right),\hfill \end{array}$$

(36)

where $R_E,R_L$ are linear resistors, $C_1,C_2$ are capacitors, E is an inductor, $R_D$ denotes the impedance of the diode, and $a>0$ is a known scalar. Moreover, $V_{C_1}\left(t\right),V_{C_2}\left(t\right)$ are the corresponding voltages at the capacitors $C_1,C_2$, respectively, $i_E\left(t\right)$ is the current through the resistance, $R_E$, and $u\left(t\right)$ is the control input voltage. The above model can be rewritten in the state space form:

$$\begin{array}{ccc}\hfill \dot{x}& =\hfill & Ax+Bu\hfill \\ \hfill y& =\hfill & Cx\hfill \end{array}$$

(37)

with

$$\begin{array}{ccc}\hfill A& =\hfill & \left[\begin{array}{ccc}-\frac{1}{R_L{C}_1}& \frac{1}{R_L{C}_1}& \frac{1}{C_1}\\ \frac{1}{R_L{C}_2}& -\frac{a}{C_2}-\frac{1}{R_L{C}_2}& 0\\ -\frac{1}{E}& 0& -\frac{R_E}{E}\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ \frac{1}{E}\end{array}\right]\hfill \\ \hfill C& =\hfill & \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right],\hfill \end{array}$$

(38)

where $x:=(x_1,x_2,x_3)$ with $x_1=V_{C_1}$, $x_2=V_{C_2}$ and $x_3=i_E$. The output $y=(x_1,x_2)$ implies that the capacitor voltages $V_{C_1}\left(t\right),V_{C_2}\left(t\right)$ can be measured while the inductor current $i_E\left(t\right)$ is not available for feedback. The parameter values in matrices A and B are given in Table 2 below.

Table 2. Parameters of the tunnel diode circuit [44].

Parameter	${\mathit{C}}_1$	${\mathit{C}}_2$	E	${\mathit{R}}_{\mathit{E}}$	${\mathit{R}}_{\mathit{L}}$	a
Value	1 F	$0.1$ F	20 H	$0\mathrm{\Omega }$	$1\mathrm{\Omega }$	$0.002$

The hybrid model (15) is developed, as described in Section 5. By checking the required conditions of Lemma 1 using the MATLAB environment with the YALMIP toolbox and SeDuMi solver [45], the LMI condition (21) is feasible with the following values: ${\epsilon }_y=2.7893$, $L=0$, $\gamma =21.2287$. By setting $\lambda =0.5$, $\tilde{\lambda }=0.6$, and $\nu =0.01$, and substituting in (28), we obtain $T=0.0256$ and $\tilde{\gamma }=638.3677$. Finally, by picking $\vartheta =0.01$, all parameters of the PETM (27) are set.

The approach is examined on a MATLAB simulation with the initial conditions $x_p(0,0)=(10,-20,20)$, $x_c(0,0)=(0,0,0)$, $e(0,0)=(0,0)$, $\eta (0,0)=0$, ${\tau }_s(0,0)=0$, ${\tau }_y(0,0)=0$ using the MATLAB toolbox HyEQ [46] for hybrid dynamical systems. By running the simulation for 20 seconds, the obtained minimum and average inter-transmission times are ${\tau }_{min}=0.0256$ and ${\tau }_{avg}=0.1394$, respectively. As expected, the minimum inter-transmission time ${\tau }_{min}$ is typically equal to the periodic sampling interval T, however, the average inter-transmission time ${\tau }_{avg}$ is larger than T, which supports our analysis and justifies the benefit of the approach compared to periodic sampling. The closed-loop response is shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

Figure 3. State trajectories of the plant and the controller.

Figure 4. State trajectories of the plant and the controller for the first two seconds.

Figure 5. Norm of the plant state.

Figure 6. Sampling-induced errors.

Figure 7. Sampling-induced errors for the first two seconds.

Figure 8. Evolution of $\eta \left(t\right)$ for the first 2 s.

Figure 9. Evolution of $\eta \left(t\right)$ for the first 2 s.

Figure 10. Evolution of $\varphi \left(t\right)$ for the first 2 s.

Figure 11. Periodic and transmission instants.

Figure 12. Periodic and transmission instants for the first 1 s.

Figure 3 shows that the plant and the controller states converge asymptotically to the origin as expected. A zoom-in of the first two seconds is illustrated in Figure 4, which shows that the observer successfully captures the plant state in a short time.

Figure 5 shows the norm of the plant state converging to the origin, affirming the asymptotic stability property as stated by the result in Theorem 1.

Figure 6 shows the evolution of the sampling-induced-errors $e\left(t\right)$, which are reset to zero at each transmission instant, as explained. A zoom-in of the first 2 s is presented in Figure 7 to clarify this behavior.

Figure 8 and Figure 9 show the trajectory of $\eta \left(t\right)$ with a zoom-in of the first two seconds, where we note that $\eta \left(t\right)>0$, as stated in Remark 2.

The trajectory of the dynamic variable $\varphi \left(t\right)$ is shown in Figure 10 for the first 2 s. It is clear that $\varphi \left(t\right)$ is strictly positive and decreases from $1/\lambda$ to $\lambda$, as discussed after (30).

The periodic time instants and the transmission instants are shown in Figure 11. We note that the transmission instants generated by the PETC are much less than the periodic sampling instants, which support the effectiveness of the proposed approach.

A zoom-in of the first 1 s for the periodic and transmission instants is shown in Figure 12 to highlight the fact that the event-triggering condition is only verified at periodic sampling instants and not in continuous-time. Note that this example has been implemented in [44] under the assumption of state feedback control, i.e., $y=x$. For the sake of comparison, we repeat the simulation with this case, and the obtained inter-transmission times are shown in Figure 13 below. From Figure 11 and Figure 13, it can be seen that the triggering instant case is much less than the output feedback control case, which implies that the output feedback control case requires more control actions as compensation due to the lack of full state information. From Figure 13, we can see that only the first triggering instant is generated with the time-triggering rule, as opposed to [44], while all the remaining instants are larger than T, which justifies the efficiency of the proposed approach.

Figure 13. Periodic and transmission instants for state feedback control.

Table 3 shows the comparison with [44] in this case.

Table 3. Comparison with [44] on the number of triggering times.

Result	No. of Periodic Samples per Second	No. of PETC Transmissions per Second
[44]	100	2
Proposed approach	10	2

From the second column of Table 3 we note that the proposed PETC produces almost the same number of triggering instants as in [44]. However, the periodic verifications of the ETC rule are reduced by 90% compared to [44]. In other words, the number of periodic instants at which the ETC needs to be checked is much lower than [44], leading to decreased energy consumption by the sensors.

Numerical Example

For the sake of comparison, we implement the proposed approach on the numerical example given in [29,38] and we compare the obtained results. The plant model in [38] is defined with

$$\begin{array}{ccc}\hfill A& =\hfill & \left[\begin{array}{cccc}1.3& -0.2& 6.7& -5.6\\ -0.6& -4.3& 0& 0.7\\ 1.1& 4.3& -6.7& 5.9\\ 0.1& 4.3& 1.3& -2.1\end{array}\right],B=\left[\begin{array}{cc}0& 0\\ 5.7& 0\\ 1.1& -3.1\\ 1.1& 0\end{array}\right]\hfill \\ \hfill C& =\hfill & \left[\begin{array}{cccc}1& 0& 1& -1\end{array}\right],\hfill \end{array}$$

(39)

We design the observer-based controller via pole placement, such that the eigenvalues of $A-BK$ are placed at $\{-1,-2,-3,-4\}$, and the eigenvalues of $A-LC$ are placed at $\{-5,-8,-11,-20\}$, yielding the following gains:

$$\begin{array}{ccc}\hfill K& =\hfill & \left[\begin{array}{cccc}-0.0916& -0.2439& 0.0336& 0.0886\\ -1.1558& -0.2701& 0.1756& -0.5371\end{array}\right]\hfill \\ \hfill L& =\hfill & {\left[\begin{array}{cccc}31.8745& 29.2897& 180.9978& 180.6723\end{array}\right]}^T.\hfill \end{array}$$

(40)

Then, by following steps similar to the previous example, we obtain ${\epsilon }_y=39.9652$, $L=0$, $\gamma =26.5843$. By setting $\lambda =0.5$, $\tilde{\lambda }=0.6$ and $\nu =0.01$, and substituting in (28), we obtain $T=0.0206$ and $\tilde{\gamma }=993.0439$. The closed-loop response is presented in Figure 14 and Figure 15.

Figure 14. State trajectories of the plant.

Figure 15. Periodic and transmission instants.

The comparison of the obtained results is presented in Table 4 below.

Table 4. Comparison with [29,38] on the number of triggering times and average transmissions.

Result	No. of PETC Transmissions	Average Inter-Transmission
	per Second	Times
[29]	603	0.0331
[38]	382	0.0523
Proposed approach	74	0.2657

The presented results in Table 4 clearly highlight the advantages of the proposed approach compared to [29,38] in terms of much less triggering instants and higher average inter-transmission intervals.

8. Design Procedure

In this section, we provide some guidelines that explain how to apply the proposed PETC approach in a systematic manner:

1.

Put the plant model in the LTI form (2).

2.

Check for controllability and observability.

3.

Define the desired eigenvalues for the controller and the observer.

4.

Compute the controller gain K and the observer gain L.

5.

Derive the hybrid model as in (15).

6.

Check the feasibility of the LMI condition (21) and find the values of PETC parameters, $T,\epsilon ,\vartheta ,\lambda ,\gamma$.

7.

Run the control system using the flow and jump sets in (15).

9. Conclusions

We studied the problem of periodic event-triggered control for linear systems based only on the output measurement. The proposed solution is well adapted to practical implementation since the event-triggering mechanism is checked only at periodic time instants rather than continuous-time verification. The problem is formulated as a hybrid dynamical system to truly describe the dynamic behavior. By using the appropriate Lyapunov function, we show that the closed-loop stability is ensured while automatically ruling out the Zeno phenomenon. The effectiveness of the approach was proven by numerical simulation. The obtained results support the analysis in this study, where the asymptotic stability is achieved, as illustrated by the state response and the trajectory of the norm of the state. The inter-sampling behavior that has been considered in the analysis is consistent with the proposed ETC implementation as the sampling error is reset to zero at each triggering instant. The fruitful inclusion of the dynamic internal variable in the PETC mechanism has led to a significant reduction in the transmission load, as reflected by the simulation comparison. We show that the required conditions to apply the proposed approach are feasible for LTI systems, as demonstrated in the electronic circuit application and the numerical example. Finally, to allow for the systematic application of the proposed method, design procedure guidelines are outlined in a logical order.

In this study, we focused on illustrating the advantage of enriching the PETC mechanism with a dynamic variable. Hence, the robustness of the approach against exogenous input, such as external disturbances, measurement noises, and transmission delays, were not considered at this stage. However, the consideration of such phenomena is relevant, and in practice, requires careful revision of the analysis, as shown in [25]. Moreover, the PETC mechanism has been developed for LTI systems, which allows casting the required conditions in terms of the feasibility of an LMI condition. Hence, the extension of the approach to nonlinear systems requires a different set of assumptions and the stability analysis needs to be thoroughly revisited. Furthermore, it is important to take into account other network-induced issues such as quantization [47], packet dropout [38], scheduling [48], and malicious attacks [49] to make the approach more adapted to practical implementation. Finally, the PETC scheme in this study has been designed by emulation, which can result in a conservative design, as discussed in [50]. Hence, the joint design for both the controller and PETC mechanism is another interesting direction for future investigations.