# Data-driven Control of Dynamic Event-triggered Systems with Delays

Xin Wang, Jian Sun, , Julian Berberich, Gang Wang, ,
Frank Allgöwer, and Jie Chen This work was supported in part by the National
Key R$\&$D Program of China under Grant 2018YFB1700100, the National Natural
Science Foundation of China under Grants 61925303, 62088101, 62173034,
U20B2073, 61720106011, and the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC 2075 -
390740016 and under grant 468094890. We acknowledge the support by the
Stuttgart Center for Simulation Science (SimTech). The authors thank the
International Max Planck Research School for Intelligent Systems (IMPRS-IS)
for supporting J. Berberich. X. Wang, J. Sun, and G. Wang are with the State
Key Lab of Intelligent Control and Decision of Complex Systems, School of
Automation, Beijing Institute of Technology, Beijing 100081, China, and also
with the Beijing Institute of Technology Chongqing Innovation Center,
Chongqing 401120, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>gangwang@bit.edu.cn). J. Berberich and F. Allgöwer are with the Institute for
Systems Theory and Automatic Control, University of Stuttgart, 70550
Stuttgart, Germany (e-mail<EMAIL_ADDRESS>frank.allgower@ist.uni-stuttgart.de). J. Chen is with the Department of
Control Science and Engineering, Tongji University, Shanghai 201804, China,
and also with the State Key Lab of Intelligent Control and Decision of Complex
Systems, School of Automation, Beijing Institute of Technology, Beijing
100081, China (e-mail: chenjie@bit.edu.cn).

###### Abstract

This paper studies data-driven control of unknown sampled-data systems with
communication delays under an event-triggering transmission mechanism. Data-
based representations for time-invariant linear systems with known or unknown
system input matrices are first developed, along with a novel class of dynamic
triggering schemes for sampled-data systems with time delays. A model-based
stability condition for the resulting event-triggered time-delay system is
established using a looped-functional approach. Combining this model-based
condition with the data-driven representations, data-based stability
conditions are derived. Building on the data-based conditions, methods for co-
designing the controller gain and the event-triggering matrix are subsequently
provided for both cases with or without using the input matrix. Finally,
numerical examples are presented to corroborate the usefulness of additional
prior knowledge of the input matrix in reducing the conservatism of stability
conditions, as well as the merits of the proposed data-driven event-triggered
control schemes relative to existing results.

###### Index Terms:

Data-driven control, event-trigger, stability, sampled-data systems

##  I Introduction

Sampled-data control has aroused great interests in the study of networked
control systems due to its convenience in system design and analysis [1, 2]. A
fundamental concept for sampled-data systems is so-called maximum sampling
interval (MSI) for maintaining system stability. In the literature, previous
approaches for analyzing sampled-data systems and computing the MSIs can be
generally categorized into four groups [3], which include the discrete-time
approach [4], impulsive systems approach [5], input/output stability approach
[6], and input-delay approach [7]. On top of the discrete-time and input-delay
approaches, a looped-functional approach was recently developed in [8, 9].
This proposal has been shown to yield (markedly) improved stability conditions
by introducing a looped-functional into the common Lyapunov functional, where
the looped-functional is required to be continuous at sampling points but not
necessarily to be positive definite; see, e.g., using a two-sided looped-
functional in [10], an integral-based looped-functional in [11], and an
extended looped-functional in [12]. In sampled-data systems, communication
delays are natural, inevitable, and they have a considerable impact on the MSI
[13, 14]. Recent efforts in [13, 15, 16] have been devoted to analyzing
sampled-data systems with communication delays by means of the looped-
functional approach.

It is worth stressing that all the aforementioned results are model-based, in
the sense that they require explicit knowledge of the underlying system
models. Nonetheless, obtaining an accurate system model is challenging and may
even be impossible in many real-world applications.  System identification
[17] can be used to determine a model based on measured data. However, it is
difficult to obtain tight error bounds in system identification when only
noisy data of finite length are available [18]. Therefore, as an alternative,
direct data-driven control without prior identification has received
increasing attention in recent years, see [19] for a survey. A number of data-
driven approaches have been put forth for different control problems using the
celebrated Fundamental Lemma [20], including state-feedback and optimal
control in [21, 22], robust control in [23, 24], predictive control in [25,
26, 27, 28]. An extension studying data-driven system stabilization of as well
as robustness guarantees for systems with time-delays and measurement noise
was recently investigated in [29]. Moreover, data-driven approaches for
estimating the MSI and designing sampled-data controllers under aperiodic
sampling were developed in [30] and [31]. Wedding the data-driven system
representation with the time-delay approach, a data-based stability condition
for _continuous-time_ sampled-data control systems was derived in [32], along
with a data-driven controller. Leveraging the stability condition, MSIs under
different levels of noise can be computed. Nonetheless, these results are
conservative due to the fact that the data-based condition in [32] follows
from the model-based one in [7]. Improved MSIs can be expected if less
conservative model-based stability conditions are employed through looped-
functional approach.

In real sampled-data systems, the digital communication network has limited
bandwidth. Reducing “redundant” data transmissions can be helpful for saving
communication resources while maintaining desired control performance. To this
aim, an event-triggering transmission strategy was proposed in [33, 34]. Its
key idea is to monitor the system continuously, but transmit sampled data only
when “necessary” according to some predefined criterion. Following this idea,
a multitude of sampling-based event-triggering schemes were devised, including
e.g., discrete event-trigger [35], dynamic event-trigger [36], dynamic
periodic event-trigger [37]. From an energy equipartition perspective, energy-
event-triggered control of cyber-physical network systems was developed in
[38] for robust and fast stabilization. However, these existing event-
triggered control methods all require explicit system models. In particular,
they cannot handle uncertainty which may arise from an identification step in
case only data are available. To the best of the authors’ knowledge, it
remains an untapped field to design data-driven event-triggering schemes as
well as controllers for unknown sampled-data control systems from noisy data
with rigorous stability guarantees.

These recent advances and developments have motivated our work in this paper,
which is focused on data-driven event-triggered control of sample-data systems
with communication delays. Motivated by dynamic periodic event-triggering from
[37], we develop a discrete-time dynamic periodic event-triggering scheme that
correlates with communication delays to save transmission resources.
Subsequently, model- and data-based stability conditions for sampled-data
systems with delays under the triggering scheme are obtained by employing the
looped-functional approach.  Based on the model-based criteria, data-driven
co-design methods of the controller gain and the triggering matrix are given
for both cases having known or unknown input matrices. Possibly large MSIs can
be obtained, similar to [32], by iteratively solving two semi-definite
programs which relax the underlying non-convex design problem. Alternative
methods using an algebraically equivalent systems [39] are then provided to
avoid such iterative computation, where the design conditions are convex in
all decision variables. Furthermore, it has been shown in [40] that leveraging
prior system knowledge could provide less conservative system analysis and
design results than purely data-driven approaches. Complementing the system
representation in [32], a new data-dependent parametrization with known input
matrices is established. It turns out that incorporating the prior knowledge
of the input matrix can help to further reduce conservatism of the resulting
data-based stability conditions, which is confirmed by our numerical results.

In words, the main contributions of this work are summarized as follows.

  1. c1)

A novel dynamic event-triggering scheme for sampled-data systems with delays,
where the dynamic threshold variable only changes at sampling points,
eliminating the burden of continuously computing the dynamic variable in [36,
37];

  2. c2)

Model- and data-based stability conditions in the form of linear matrix
inequalities (LMIs) for event-triggered control systems with delays based on a
novel looped-functional; and,

  3. c3)

Data-driven approaches to co-designing the controller gain and the triggering
matrix based on a data-based system parametrization for both cases with known
and unknown input matrices.

Practical merits and effectiveness of the proposed data-driven event-triggered
control as well as stability conditions are finally demonstrated through
numerical comparison using two simulated examples.

The remainder of this paper is structured as follows. In Section II, we
introduce data-based system representations, as well as put forth a dynamic
event-triggering scheme for sampled-data systems with noise and time delays.
Based on the results in Section II, both model- and data-based stability
conditions are developed in Section III. Subsequently, methods for co-
designing the controller gain and the triggering matrix are discussed. Section
IV validates the merits and application of our methods and conditions on
practical systems. Finally, Section V draws concluding remarks.

Notation. Throughout this paper, $\mathbb{N}$, $\mathbb{R^{+}}$,
$\mathbb{R}^{n}$, and $\mathbb{R}^{n\times m}$ denote the sets of all non-
negative integers, non-negative real numbers, $n$-dimensional vectors, and
${n\times m}$ real matrices, respectively. Symbol $P\succ 0$ ($P\succeq 0$)
means $P$ is a symmetric positive (semi)definite matrix; ${\rm
diag}\\{\cdots\\}$ denotes a block-diagonal matrix; ${\rm Sym}\\{P\\}$
represents the sum of $P^{T}$ and $P$. Let ${\rm col}\\{\cdots\\}$ represent
column vectors; $0$ and $I$ stand for zero and identity matrices of
appropriate dimensions. Notation ‘$\ast$’ denotes the symmetric term in
symmetric block matrices; We use $\|\cdot\|$ to denote the Euclidean norm of a
vector. $t_{k}^{-}$ means that $t$ approaches $t_{k}$ from left.
$\\{\tau_{j}-d\\}_{j}$ represents the set
$\\{\tau_{0}-d,\tau_{1}-d,\tau_{2}-d,\dots\\}$. We write $[\cdot]$ if elements
in the matrix can be inferred by symmetry.

##  II Preliminaries

###  II-A Sampled-data control with unknown system matrices

Consider the following linear time-invariant system

$\dot{x}(t)=Ax(t)+Bu(t)$ (1)

where $t\geq 0$, $x(t)\in\mathbb{R}^{n}$ is the system state, and
$u(t)\in\mathbb{R}^{m}$ is the control input. $A\in\mathbb{R}^{n\times n}$ and
$B\in\mathbb{R}^{n\times m}$ are constant system matrices. An _unknown_
constant time-delay $d$ in the network communication is considered and it
satisfies $0\leq\underline{d}\leq d\leq\bar{d}$, where the bounds
$\underline{d}$ and $\bar{d}$ are given constants. We assume that some pre-
collected state-input measurements are available. Then two different
assumptions on system matrices $A$ and $B$ are considered in this paper, which
are formalized as follows.

######  Assumption 1.

The matrices $A$ and $B$ are unknown.

######  Assumption 2.

The matrix $A$ is unknown, and $B$ is known.

Our main goal is to design a fixed control gain matrix
$K\in\mathbb{R}^{m\times n}$ such that the state-feedback control signal
$u(t)=Kx(t-d)$ stabilizes system (1) under Assumption 1 or 2. Suppose that the
control signals are released at discrete time instants

$\displaystyle 0\leq d=t_{0}<t_{1}<\dots<t_{k}<\dots<\infty.$ (2)

The sampled-data system in closed loop can be written as

$\left\\{\begin{array}[]{lll}\dot{x}(t)=Ax(t)&,&t\in[0,t_{0})\\\
\dot{x}(t)=Ax(t)+BKx(t_{k}-d)&,&t\in[t_{k},t_{k+1}).\\\
x(t)=0&,&t\in[-d,0)\end{array}\right.$ (3)

A framework for system (3) is presented in Fig. 1. Two differences relative to
classic sampled-data systems can be noted: i) including a dynamic event-
triggering transmission module to determine the transmission sequence
$\\{t_{k}\\}_{k\in\mathbb{N}}$, and ii) a data-based system representation for
co-designing the event-triggering scheme and the controller. The data-based
system representation is constructed based on the data collected offline at
discrete instants $\\{T_{i}\\}_{i}$. These sampling times are independent of
the closed-loop sampling times $\\{t_{k}\\}_{k}$ at which the controller is
updated. Next, we elaborate on the data-based representation and the designed
event-triggering transmission scheme, separately.

Figure 1: Graphical illustration of the proposed approach.

###  II-B Data-based system representation with noise

Under Assumption 1, a main challenge is that both system stability analysis
and controller design should be performed without knowledge of the system
matrices $A$ and $B$. To this end, we recall the data-based representation
developed in [24, 32] for system (1).  Suppose that sampled measurements
$\\{\dot{x}(T_{i}),\,x(T_{i}),\,u(T_{i})\\}^{\rho}_{i=1}$
$(i,\,\rho\in\mathbb{N})$ of the perturbed system

$\dot{x}(t)=Ax(t)+Bu(t)+B_{w}w(t)$ (4)

are available at discrete time instants $T_{i}\in[0,t]$, where
$B_{w}\in\mathbb{R}^{n\times n_{w}}$ is assumed known and has full column
rank. This is without loss of generality since one can also lump $B_{w}$ into
the additive noise $w(t)$. Here, the measured system data are corrupted by the
unknown noise (perturbation) sequence $\\{w(T_{i})\\}^{\rho}_{i=1}$, where
$w(t)\in\mathbb{R}^{n_{w}}$ captures the unknown noise or the unmodeled system
dynamics. These measurements can be collected to form the following data
matrices

$\displaystyle\dot{X}$
$\displaystyle:=[\dot{x}(T_{1})~{}~{}\dot{x}(T_{2})~{}~{}\dots~{}~{}\dot{x}(T_{\rho})]$
$\displaystyle X$
$\displaystyle:=[x(T_{1})~{}~{}x(T_{2})~{}~{}\dots~{}~{}x(T_{\rho})]$
$\displaystyle U$
$\displaystyle:=[u(T_{1})~{}~{}u(T_{2})~{}~{}\dots~{}~{}u(T_{\rho})]$
$\displaystyle W$
$\displaystyle:=[w(T_{1})~{}~{}w(T_{2})~{}~{}\dots~{}~{}w(T_{\rho})]$

where $\dot{X}$, $X$, and $U$ are available while $W$ is unknown. Then, it is
evident that

$\displaystyle\dot{X}=AX+BU+B_{w}W.$ (5)

In practice, the noise is typically bounded. We make a standing assumption on
the noise that was also used in [24] and [32].

######  Assumption 3 (_Noise bound_).

The noise sequence $\\{w(T_{i})\\}^{\rho}_{i=1}$ collected in the matrix $W$
satisfies $W\in\mathcal{W}$ with

$\displaystyle\mathcal{W}=\bigg{\\{}W\in\mathbb{R}^{n_{w}\times\rho}\Big{|}\left[\\!\begin{array}[]{cc}W^{T}\\\
I\\\ \end{array}\\!\right]^{T}\left[\\!\begin{array}[]{cc}Q_{d}\\!&\\!S_{d}\\\
\ast\\!&\\!R_{d}\\\ \end{array}\\!\right]\left[\\!\begin{array}[]{cc}W^{T}\\\
I\\\ \end{array}\\!\right]\succeq 0\bigg{\\}}$ (12)

for some known matrices $Q_{d}\prec 0\in\mathbb{R}^{\rho\times\rho}$,
$S_{d}\in\mathbb{R}^{\rho\times n_{w}}$, and
$R_{d}=R_{d}^{T}\in\mathbb{R}^{n_{w}\times n_{w}}$.

######  Remark 1 (_Explanation of noise_).

Assumption 3 provides a general form for modeling bounded additive noise,
which has been used in similar form by [21, 32, 24, 29]. The matrices $Q_{d}$,
$S_{d}$, $R_{d}$ are chosen depending on the prior knowledge on the noise. For
example, if $w(t)\in[-\bar{w},\bar{w}]$ for $\bar{w}\geq 0$, (12) boils down
to

$\displaystyle
WW^{T}=\sum\limits_{i=1}^{i=\rho}w(T_{i})w^{T}(T_{i})\preceq\bar{w}^{2}\rho I$
(13)

where $Q_{d}=-I$, $R_{d}=0$, and $R_{d}=\bar{w}^{2}\rho I$. The noise bound in
Assumption 12 has been employed in many recent works on data-driven control,
and allows for various extensions (e.g., using multipliers instead of constant
matrices $Q_{d}$, $S_{d}$, and $R_{d}$ in [40]).

Under the model in (5) and Assumption 3, we define $\Sigma_{s}$ as the set of
all pairs $(A~{}B)$ adhering to the measured data and the noise bound, namely

$\displaystyle\Sigma_{s}:=\\{(A~{}B)\mid\dot{X}=AX+BU+B_{w}W,~{}W\in\mathcal{W}\\}.$
(14)

Through a simple mathematical replacement method, an equivalent expression of
$\Sigma_{s}$ in the form of a quadratic matrix inequality (cf. [32, Theorem
1]) is given below.

######  Lemma 1 (_Data-based representation_).

The set $\Sigma_{s}$ in (14) is equivalent to

$\displaystyle\Sigma_{s}=\bigg{\\{}[A~{}B]\in\mathbb{R}^{\color[rgb]{0,0,1}n\times(n+m)}\Big{|}\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\
I\\\
\end{array}\\!\\!\\!\right]^{T}\\!\Theta_{s}\\!\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\
I\\\ \end{array}\\!\\!\\!\right]\succeq 0\bigg{\\}}$ (19)

where

$\displaystyle\Theta_{s}=\left[\\!\begin{array}[]{cc}Q_{c}\\!&\\!S_{c}\\\
\ast\\!&\\!R_{c}\\\
\end{array}\\!\right]:=\left[\\!\begin{array}[]{cc}-\mathcal{Z}\\!&\\!0\\\
\dot{X}\\!&\\!B_{w}\\\
\end{array}\\!\right]\left[\\!\begin{array}[]{cc}Q_{d}\\!&\\!S_{d}\\\
\ast\\!&\\!R_{d}\\\
\end{array}\\!\right]\left[\\!\begin{array}[]{cc}-\mathcal{Z}\\!&\\!0\\\
\dot{X}\\!&\\!B_{w}\\\ \end{array}\\!\right]^{T}$ (28)

with $\mathcal{Z}:={\rm col}\\{X,\,U\\}$.

For the sake of stability analysis in subsequent sections, a technical
assumption similar to [32, Assumption 3] on the matrix $\Theta_{s}$ in Lemma 1
is made below.

######  Assumption 4.

The matrix $\Theta_{s}$ is invertible and has $n_{w}$ positive eigenvalues.

Lemma 1 provides a purely data-based representation of system (1) with unknown
matrices $A$ and $B$ (cf. Assumption 1) using noisy data $\dot{X}$, $X$, and
$U$. If additional prior knowledge on the system is given, then conservatism
can be reduced in the co-design step of controller and triggering mechanism.
Specifically, we consider the case of knowing the input matrix $B$ (cf.
Assumption 2), which allows us to state significantly less conservative design
conditions. To treat the data-driven control under Assumption 2, the set of
all matrices $A$ consistent with the model in (5) and the noise bound in
Assumption 3 is defined as follows.

$\displaystyle\bar{\Sigma}_{s}:=\\{A\in\mathbb{R}^{n\times
n}\mid\dot{X}=AX+BU+B_{w}W,~{}W\in\mathcal{W}\\}.$ (29)

Similar to Lemma 1, an equivalent representation for $\bar{\Sigma}_{s}$ is
given in the following.

######  Lemma 2 (Data-based representation with known input matrix $B$).

The set in (29) is equal to

$\displaystyle\bar{\Sigma}_{s}=\bigg{\\{}A\in\mathbb{R}^{n\times
n}\Big{|}\left[\begin{array}[]{cc}A^{T}\\\ I\\\
\end{array}\right]^{T}\bar{\Theta}_{s}\left[\begin{array}[]{cc}A^{T}\\\ I\\\
\end{array}\right]\succeq 0\bigg{\\}}$ (34)

where

$\displaystyle\bar{\Theta}_{s}=\left[\\!\begin{array}[]{cc}\bar{Q}_{c}\\!&\\!\bar{S}_{c}\\\
\ast\\!&\\!\bar{R}_{c}\\\
\end{array}\\!\right]:=\left[\\!\begin{array}[]{cc}-X\\!&\\!0\\\
\mathcal{U}\\!&\\!B_{w}\\\
\end{array}\\!\right]\left[\\!\begin{array}[]{cc}Q_{d}\\!&\\!S_{d}\\\
\ast\\!&\\!R_{d}\\\
\end{array}\\!\right]\left[\\!\begin{array}[]{cc}-X\\!&\\!0\\\
\mathcal{U}\\!&\\!B_{w}\\\ \end{array}\\!\right]^{T}$ (43)

with $\mathcal{U}:=\dot{X}-BU$.

Lemma 2 presents a data-based representation of system (1) while accounting
for prior knowledge of the input matrix $B$. As discussed in [40], additional
prior knowledge of the system may help shrink the set of systems consistent
with the collected data, thereby contributing to less conservative data-based
stability conditions compared to the ones derived using only data. This can be
understood as a data- and model-driven hybrid approach. In fact, it can be
shown that the set $\bar{\Sigma}_{s}$ in Lemma 2 is tighter than $\Sigma_{s}$
in Lemma 1 for system (1). On the other hand, prior knowledge of the input
matrix $B$ may not always be available in general. These considerations thus
motivates us to pursue data-driven control schemes based on $(i)$ pure data as
well as $(ii)$ data along with prior knowledge of $B$, which are detailed in
Section III.

######  Assumption 5.

The matrix $\bar{\Theta}_{s}$ is invertible and has $n_{w}$ positive
eigenvalues.

In practice, Assumptions 4 and 5 hold if that the available measurements of
the system states are sufficiently rich. To be precise, Assumptions 4 and 5
hold for the common special case $S_{d}=0$ if i) $\mathcal{Z}$ (Assumption 4)
or $X$ (Assumption 5) has full row rank, ii) $B_{w}$ is invertible, i.e.,
$n_{w}=n$, and iii) the disturbance $W$ generating the data satisfies a strict
version of the quadratic matrix inequality in (12), i.e.,
$WQ_{d}W^{T}+R_{d}\succ 0$. The proof has been given in [32, Appendix B].

######  Remark 2 (_Estimation of state derivatives_).

In practice, the time derivatives $\dot{X}$ are typically not available.
However, by means of finite differences and with dense recordings of the
system states $X$, a procedure for estimating the state derivatives has been
discussed in [32]. Specifically, one can approximate $\dot{x}(T_{i})$ by
$\underline{\dot{x}}(T_{i})=\frac{x(T_{i+1})-x(T_{i})}{T_{i+1}-T_{i}}$ with
error bound
$\|\underline{\dot{x}}(T_{i})-\dot{x}(T_{i})\|_{2}\leq\frac{\bar{a}(T_{i+1}-T_{i})}{2}\Big{[}\bar{a}\|x(T_{i})\|_{2}+(1+\frac{\bar{a}(T_{i+1}-T_{i})}{3}\bar{b})\|u(T_{i})\|_{2}\Big{]}$,
where $\bar{a}$ and $\bar{b}$ are known bounds for $\|A\|_{2}\leq\bar{a}$ and
$\|B\|_{2}\leq\bar{b}$. Alternatively, several derivative estimation methods
based on continuous-time system identification can be found in [41]. In this
paper, we focus on design and analysis of data-driven event-triggered control,
where data of the derivative are available, but exploring further
possibilities to estimate $\dot{X}$ is an interesting issue for future
research.

######  Remark 3 (_Related work_).

Lemma 1 from [32] contributes a purely data-based representation for sets of
continuous-time systems using only some pre-collected input-state data.
Compared to other approaches for data-driven control of continuous-time
systems, e.g. in [21], Lemma 1 enjoys reduced computational complexity as well
as enhanced robustness against additive noise. Besides, thanks to its form of
linear matrix inequalities, the data-based representation in (19) can be
naturally married with model-based sampled-data control methods, such as the
discrete-time approach, input-delay approach, input-output approach, and
impulsive systems approach, to study data-driven control of sampled-data
systems; see, e.g., [32, 30, 31]. Overall, the work of [32] put forward a
unifying data-driven control framework for continuous-time sampled-data
systems with time delays. In this paper, we build on Lemmas 1 and 2 to
investigate data-driven control of event-triggered sampled-data systems with
delays by developing a data-driven looped-functional analysis.

###  II-C Dynamic event-triggering transmission scheme

In the sampled-data control system in Fig. 1, a dynamic event-triggering
module is employed to dictate the transmission instants
$\\{t_{k}\\}_{k\in\mathbb{N}}$. In contrast to classical time-triggered
transmission schemes, _only_ “informative” measurements are transmitted by
event-triggered schemes to save transmission resources. Concretely, our
proposed dynamic event-triggering scheme consists of two steps. The first
step, similar to _time-triggering scheme_ , is to sample the system state at
time instants $\\{t_{k}+ih-d\\}_{i,k\in\mathbb{N}}$, where the sampling
interval $h$ satisfies $0<\underline{h}\leq h\leq\bar{h}$. In the second step,
that is _event-triggering scheme_ , the sampled signal $x(t_{k}+ih-d)$ is to
be checked for the following condition

$\displaystyle\eta(\tau_{j}-d)+\theta\rho(\tau_{j}-d)<0,$ (44)

where $\rho$ is a discrete-time function
$\rho:\\{\tau_{j}-d\\}_{j}\rightarrow\mathbb{R}$, defined as follows

$\displaystyle\rho:=$ $\displaystyle~{}\sigma_{1}x^{T}(\tau_{j}-d)\Omega
x(\tau_{j}-d)-e^{T}(\tau_{j}-d)\Omega e(\tau_{j}-d)$ (45)
$\displaystyle+\sigma_{2}x^{T}(t_{k}-d)\Omega x(t_{k}-d),$

where $\tau_{j}=t_{k}+jh$, $j=0,1,\cdots,m_{k}$, with
$m_{k}=\frac{t_{k+1}-t_{k}}{h}-1$; $\Omega\succeq 0$ is some weight matrix;
$\sigma_{1}$, $\sigma_{2}$, and $\theta\geq 0$ are parameters to be designed;
$e(\tau_{j}-d):=x(t_{k}-d)-x(\tau_{j}-d)$ denotes the error between sampled
signals $x(t_{k}-d)$ at the latest transmission instant and $x(\tau_{j}-d)$ at
the current sampling instant; and, $\eta(\tau_{j}-d)$ is a dynamically
evolving parameter $\eta:\\{\tau_{j}-d\\}_{j}\rightarrow\mathbb{R}$,
satisfying the following difference equation

$\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)=-\lambda\eta(\tau_{j}-d)+\rho(\tau_{j}-d)$
(46)

for which $\eta(0)\geq 0$ and $\lambda>0$ are given.

If the condition (44) is met, the current sampled state $x(\tau_{j}-d)$ is
transmitted to the controller, and a zero-order holder (ZOH) follows to
maintain it within the interval $[t_{k},t_{k+1})$. Then, the triggering
algorithm is updated and detects the next sampled datum. In summary, the
dynamic event triggering condition can be written as follows

$\displaystyle
t_{k+1}=t_{k}+h\min_{j\in\mathbb{N}}\Big{\\{}j>0\Big{|}\eta(\tau_{j}-d)+\theta\rho(\tau_{j}-d)<0\Big{\\}}.$
(47)

In Fig. 2, a scheme illustrating the periodic sampling with the proposed
triggering transmission scheme is provided. The system states are sampled
periodically. Then, the delayed sampled data are transmitted to the controller
and arrive at $\\{t_{k}\\}_{k}$, if the condition (44) is satisfied, i.e., at
time instants $t_{1}-d$, $t_{2}-d$, and $t_{3}-d$.

Figure 2: Evolution of sampling and transmission events.

The following result states that the value of $\eta(\tau_{j}-d)$ remains
nonnegative for system (3) under the triggering condition (47) with $t\geq 0$
if the initial value $\eta(0)\geq 0$.

######  Lemma 3 (_Non-negativity_).

Let $\eta(0)\geq 0$, $\lambda>0$ be positive constants,
$\Omega=\Omega^{T}\succeq 0$ be a positive semi-definite matrix, and
$\theta>0$ be a positive constant satisfying $1-\lambda-\frac{1}{\theta}\geq
0$, or $\theta=0$. Then, it holds that $\eta(\tau_{j}-d)\geq 0$, for all
$j\in\mathbb{N}$ under the triggering condition in (47).

###### Proof.

According to the event-triggering condition in (44) for $t\geq 0$, the
condition is not met, which ensures that

$\displaystyle\eta(\tau_{j}-d)+\theta\rho(\tau_{j}-d)\geq 0,~{}\forall
j\in\mathbb{N}.$ (48)

If $\theta=0$, it obviously holds that $\eta(\tau_{j}-d)\geq 0$. Hence, let us
focus on $\theta>0$. Inequality (48) suggests that

$\displaystyle\rho(\tau_{j}-d)\geq-\frac{1}{\theta}\eta(\tau_{j}-d).$ (49)

From (46), it is obtained that

$\displaystyle\eta(\tau_{j+1}-d)\geq(1-\lambda)\eta(\tau_{j}-d)-\frac{1}{\theta}\eta(\tau_{j}-d).$
(50)

By induction along with the initial condition $\eta(0)\geq 0$ and the
assumption $1-\lambda-\frac{1}{\theta}\geq 0$, it follows that
$\eta(\tau_{j}-d)\geq 0$, for all $j\in\mathbb{N}$, completing the proof. ∎

######  Remark 4 (_Time delay_).

Time-delays [13, 14, 42, 15] are frequently introduced into control loops when
implementing practical control systems through communication networks (see the
structure in Fig. 1); e.g., constant communication delays in the sensor-to-
controller channel have been considered in event-triggered systems [39] and
sampled-data systems [15]. Time delays can endanger stability properties of
the closed loop. Thus, an important objective of stability analysis is to find
an admissible delay region $[\underline{d},\bar{d}]$. In this paper, we assume
that the upper bound $\bar{d}$ of the constant delay in the sensor-to-
controller channel satisfies $\bar{d}\in[0,h)$. Therefore, it is clear that
$t_{k}<t_{k+1}$, which implies that no disordered transmission packets occur.
For simplicity, the computation delays in the controller and sensor are
neglected.

######  Remark 5 (_Comparison of different event-triggering schemes_).

The event-triggering scheme (47) can be seen as a discrete-time generalization
of the continuous dynamic periodic one studied in [37]. Different from [37],
the dynamic variable $\eta(t)$ only changes at discrete sampling points
$\tau_{j}$, which eliminates the computational burden caused by continuously
computing $\eta(t)$ in [37]. Besides, our proposed scheme subsumes several
existing ones as special cases. For instance, if $d=0$, $\sigma_{1}=0$, and
$\theta$ goes to $\infty$, the condition (47) becomes the discrete event-
triggering scheme in [35]. By setting $d=0$, $\sigma_{1}=0$, and $h=0$, our
scheme boils down to the dynamic event-triggering scheme in [36]; and, it also
degenerates to a periodic transmission scheme when $d=0$, $\sigma_{1}=0$,
$\sigma_{2}=0$, and $\theta$ goes to $\infty$. Thus, our triggering scheme
unifies existing event-triggering schemes, and is expected to save
transmission resources.

##  III Main Results

In this section, we analyze the stability of system (3) under the transmission
scheme in (47), and derive an upper bound on the sampling interval $h$. A
data-based stability condition is then reproduced from the model-based
condition, and used for co-designing the triggering matrix $\Omega$ as well as
the controller gain $K$ from data.

######  Lemma 4.

For any vectors $\vartheta\in\mathbb{R}^{m}$, matrices
$R=R^{T}\in\mathbb{R}^{n\times n}\succ 0$, $N\in\mathbb{R}^{m\times 2n}$,
scalars $\alpha\leq\beta\in\mathbb{N}$, and a differentiable function
$x:[\alpha,\,\beta]\rightarrow\mathbb{R}^{n}$, the following integral
inequalities hold true

$\displaystyle-\int_{\alpha}^{\beta}\dot{x}^{T}(s)R\dot{x}(s)ds\leq$
$\displaystyle(\beta-\alpha)\vartheta^{T}N\mathcal{R}^{-1}N^{T}\vartheta$
$\displaystyle+{\rm Sym}\left\\{\vartheta^{T}N\Pi\right\\}$

where

$\displaystyle\mathcal{R}$ $\displaystyle:={\rm diag}\left\\{R,\,3R\right\\},$
$\displaystyle\Pi$
$\displaystyle:=\left[x^{T}(\beta)\\!-\\!x^{T}(\alpha),x^{T}(\beta)\\!+\\!x^{T}(\alpha)\\!-\\!2\int_{\alpha}^{\beta}\frac{x^{T}(s)}{\beta-\alpha}ds\right]^{T}.$

Lemma 4 can be cast as a special case of [42, Lemma 1], whose proof is omitted
here. Lemma 4 together with the looped-functional approach [8, Theorem 1] are
crucial ingredients of our theoretical analysis.

###  III-A Model-based stability analysis

We begin by developing a model-based stability analysis of the event-triggered
system (3) in this subsection, where matrices $A$ and $B$ are firstly assumed
_known_. The proposed triggering strategy (47) comprises a step of time-
trigger with sampling interval $h$. A bound on the MSI $h$ can be computed
through the following stability condition.

######  Theorem 1 (_Model-based condition_).

For given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$ satisfying
$1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, system (3) is
asymptotically stable under the triggering condition in (47), and
$\eta(\tau_{j}-d)$ converges to the origin for any $\eta(0)\geq 0$, if there
exist matrices $P\succ 0$, $Z\succ 0$, $T\succ 0$, $R_{1}\succ 0$, $R_{2}\succ
0$, $\Omega\succ 0$, $S=S^{T}$, $N$, $M_{1}$, $M_{2}$, $F$, such that the
following LMIs hold for all $h\in\\{\underline{h},\bar{h}\\}$ and
$d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\begin{array}[]{ccc}\Xi_{0}+h\Xi_{a}+\Psi+\mathcal{O}&dN\\\
\ast&d\mathcal{T}\end{array}\right]\prec 0$ (53)
$\displaystyle\left[\begin{array}[]{ccc}\Xi_{0}+h\Xi_{b}+\Psi+\mathcal{O}&dN&h\mathcal{M}\\\
\ast&d\mathcal{T}&0\\\ \ast&\ast&h\mathcal{R}\end{array}\right]\prec 0$ (57)

where

$\displaystyle\Xi_{0}:=$ $\displaystyle~{}{\rm
Sym}\Big{\\{}{\Pi}_{1}^{T}P{\Pi}_{3}+d{\Pi}_{2}^{T}P{\Pi}_{3}+N{\Pi}_{6}+M_{1}\Pi_{8}+M_{2}\Pi_{9}\Big{\\}}$
$\displaystyle+L_{3}^{T}\mathcal{T}L_{3}+{\Pi}_{4}^{T}Z{\Pi}_{4}-{\Pi}_{5}^{T}Z{\Pi}_{5},$
$\displaystyle\Xi_{a}:=$
$\displaystyle~{}\Pi_{7}^{T}S\Pi_{7}+L_{3}^{T}R_{1}L_{3}+L_{4}^{T}R_{2}L_{4},~{}\Xi_{b}:=-\Pi_{7}^{T}S\Pi_{7},$
$\displaystyle\Psi:=$ $\displaystyle~{}{\rm
Sym}\big{\\{}F(AL_{1}+BL_{10}-L_{3})\big{\\}},$ $\displaystyle\mathcal{O}:=$
$\displaystyle~{}\sigma_{1}L_{7}^{T}\Omega L_{7}+\sigma_{2}L_{10}^{T}\Omega
L_{10}-(L_{7}-L_{10})^{T}\Omega(L_{7}-L_{10}),$ $\displaystyle\mathcal{R}:=$
$\displaystyle~{}{\rm
diag}\left\\{-R_{1},\,-3R_{1},\,-R_{2},\,-3R_{2}\right\\},$
$\displaystyle\mathcal{T}:=$ $\displaystyle~{}{\rm
diag}\left\\{-T,\,-3T\right\\},~{}~{}~{}~{}\mathcal{M}:=\left[M_{1}~{}M_{2}\right],$
$\displaystyle{\Pi}_{1}:=$
$\displaystyle~{}[L_{1}^{T},\,L_{2}^{T},\,L_{0}^{T}]^{T},~{}~{}~{}~{}~{}~{}\,{\Pi}_{2}:=[L_{0}^{T},\,L_{0}^{T},\,L_{4}^{T}]^{T},$
$\displaystyle{\Pi}_{3}:=$
$\displaystyle~{}[l_{3}^{T},\,L_{4},\,L_{1}^{T}-L_{2}^{T}]^{T},~{}{\Pi}_{4}:=[L_{1}^{T},\,L_{3}^{T}],$
$\displaystyle{\Pi}_{5}:=$
$\displaystyle~{}[L_{2}^{T},\,L_{4}^{T}],~{}~{}{\Pi}_{6}:=[L_{1}^{T}-L_{2}^{T},\,L_{1}^{T}+L_{2}^{T}-2L_{5}^{T}]^{T},$
$\displaystyle\Pi_{7}:=$
$\displaystyle~{}[L_{6}^{T},\,L_{7}^{T}],~{}~{}\Pi_{8}:=[L_{1}^{T}-L_{6}^{T},\,L_{1}^{T}+L_{6}^{T}-2L_{8}^{T}]^{T},$
$\displaystyle\Pi_{9}:=$
$\displaystyle~{}[L_{2}^{T}-L_{7}^{T},\,L_{2}^{T}+L_{7}^{T}-2L_{9}^{T}]^{T},$
$\displaystyle L_{i}:=$
$\displaystyle~{}\left[0_{n\times(i-1)n},\,I_{n},\,0_{n\times(10-i)n}\right],\;i=1,\ldots,10,$
$\displaystyle L_{0}:=$ $\displaystyle~{}0_{n\times 10n}.$

###### Proof.

Considering the interval $[\tau_{j},\tau_{j+1})$ that satisfies
$[t_{k},\,t_{k+1})=\bigcup\limits_{j=0}^{j=m}[\tau_{j},\tau_{j+1})$, we choose
the following functional for system (3)

$\displaystyle{}{\color[rgb]{0,0,1}V(x,t)=V_{a}(t)+V_{l}(t),~{}t\in[\tau_{j},\tau_{j+1})}$
(58)

where

$\displaystyle V_{a}(t)=$
$\displaystyle~{}\big{[}x^{T}(t)~{}x^{T}(t-d)~{}\int_{t-d}^{t}x^{T}(s)ds\big{]}P[\cdot]^{T}$
$\displaystyle+\int_{t-d}^{t}\phi^{T}(s)Z\phi(s)ds+\int_{t-d}^{t}\int_{s}^{t}\dot{x}^{T}(\upsilon)T{\color[rgb]{0,0,1}\dot{x}(\upsilon)}d\upsilon\,ds$
(59)

with $\phi(s):=[x^{T}(s),\,\dot{x}^{T}(s)]^{T}$, $P\succ 0$, $Z\succ 0$, and
$T\succ 0$; and, a looped-functional $V_{l}(t)$ is constructed as follows

$\displaystyle V_{l}(t)=$
$\displaystyle~{}(\tau_{j+1}-t)(t-\tau_{j})\big{[}x^{T}(\tau_{j})~{}x^{T}(\tau_{j}-d)\big{]}S[\cdot]^{T}$
(60)
$\displaystyle+(\tau_{j+1}-t)\int_{\tau_{j}}^{t}\dot{x}^{T}(s)R_{1}\dot{x}(s)ds$
$\displaystyle+(\tau_{j+1}-t)\int_{\tau_{j}-d}^{t-d}\dot{x}^{T}(s)R_{2}\dot{x}(s)ds$

where $S=S^{T}$, $R_{1}\succ 0$, and $R_{2}\succ 0$ are to be designed. The
functional $V_{a}(t)$ is positive definite, while $V_{l}(t)$ satisfies
$\lim\limits_{t\rightarrow\tau_{j}}V_{l}(t)=\lim\limits_{t\rightarrow\tau_{j+1}}V_{l}(t)$.
Following the loop-functional approach [8], we have the next steps.

Taking the derivative of $V(t)$ along the trajectory of system (3) yields

$\displaystyle{}\dot{V}(t)=\frac{d}{dt}\big{[}V_{a}(t)+V_{l}(t)\big{]},$ (61)

where $\dot{\eta}(t)$ is given in (46), and

$\displaystyle\dot{V}_{a}(t)=\xi^{T}(t)\Big{[}$ $\displaystyle
2({\Pi}_{1}^{T}P{\Pi}_{3}+d{\Pi}_{2}^{T}P{\Pi}_{3})+L_{3}^{T}TL_{3}+{\Pi}_{4}^{T}Z{\Pi}_{4}$
$\displaystyle-{\Pi}_{5}^{T}Z{\Pi}_{5}\Big{]}\xi(t)-\int_{t-d}^{t}\dot{x}^{T}(s)T\dot{x}(s)ds,$
$\displaystyle\dot{V}_{l}(t)=\xi^{T}(t)\Big{[}$
$\displaystyle(\tau_{j+1}-t)\big{(}\Pi_{7}^{T}S\Pi_{7}+L_{3}^{T}R_{1}L_{3}+L_{4}^{T}R_{2}L_{4}\big{)}$
$\displaystyle-(t-\tau_{j})\Pi_{7}^{T}S\Pi_{7}\Big{]}\xi(t)-\\!\int_{\tau_{j}}^{t}\dot{x}^{T}(s)R_{1}\dot{x}(s)ds$
$\displaystyle-\int_{\tau_{j}-d}^{t-d}\dot{x}^{T}(s)R_{2}\dot{x}(s)ds.$

where the notation $\xi(t)$ is given as

$\displaystyle\xi(t):=\bigg{[}x^{T}(t),x^{T}(t-d),\dot{x}^{T}(t),\dot{x}^{T}(t-d),\int_{t-d}^{t}\frac{x(s)}{d}ds,$
$\displaystyle
x^{T}(\tau_{j}),\\!x^{T}(\tau_{j}\\!-\\!d),\\!\int_{t}^{\tau_{j}}\\!\frac{x^{T}(s)}{t\\!-\\!\tau_{j}}ds,\\!\int_{t\\!-\\!d}^{\tau_{j}\\!-\\!d}\\!\frac{x^{T}(s)}{t-\tau_{j}}ds,\\!x^{T}(t_{k}\\!-\\!d)\\!\bigg{]}^{T}$

Note that there are some integral terms present in $\dot{V}(t)$. According to
Lemma 4, those terms satisfy that

$\displaystyle-\int_{t-d}^{t}\dot{x}^{T}(s)T\dot{x}(s)ds\leq\xi^{T}(t)\Big{(}dN{\mathcal{T}}^{-1}N^{T}+2N{\Pi}_{6}\Big{)}\xi(t)$
(62)
$\displaystyle-\int_{\tau_{j}}^{t}\dot{x}^{T}(s)R_{1}\dot{x}(s)ds-\int_{\tau_{j}-d}^{t-d}\dot{x}^{T}(s)R_{1}\dot{x}(s)ds$
$\displaystyle~{}\leq\xi^{T}(t)\Big{(}(t-\tau_{j})\mathcal{M}\mathcal{R}^{-1}\mathcal{M}^{T}+2M_{1}\Pi_{8}+2M_{2}\Pi_{9}\Big{)}\xi(t)$
(63)

where $N$, $M_{1}$ and $M_{2}$ are suitable matrices.

According to (3), it is clear that for $t\in[\tau_{j},\tau_{j+1})$

$\displaystyle{}0$
$\displaystyle=2\xi^{T}(t)F\big{[}Ax(t)+BKx(t_{k}-d)-\dot{x}(t)\big{]}$
$\displaystyle=2\xi^{T}(t)F\big{(}Al_{1}+BKL_{10}-L_{3}\big{)}\xi(t)$ (64)

where $F$ is a matrix of dimension $10n\times n$.

Summing up (61)—(III-A) gives rise to

$\displaystyle\dot{V}(t)\leq\xi^{T}(t)\Big{[}$
$\displaystyle(\tau_{j+1}-t)\Xi_{a}+(t-\tau_{j})(\Xi_{b}+\mathcal{M}\mathcal{R}^{-1}\mathcal{M}^{T})$
(65) $\displaystyle+\Xi_{0}+\Psi+dN{\mathcal{T}}^{-1}N^{T}\Big{]}\xi(t).$

In light of the triggering condition (47) for $t\in[t_{k},\,t_{k+1})$, Lemma 3
asserts that $\eta(\tau_{j}-d)\geq 0$ for $\eta(0)\geq 0$, $\lambda>0$, and
$\theta>0$ satisfying $1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$.
Besides, according to the equation (46), it holds that

$\displaystyle\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)=-\lambda\eta(\tau_{j}-d)+\rho(\tau_{j}-d).$

From (45), $\rho(\tau_{j}-d)$ is written to be
$\rho(\tau_{j}-d)=\xi^{T}(t)\mathcal{O}\xi(t)$. Thus, it easy to be derived
that

$\displaystyle\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)\leq\xi^{T}(t)\mathcal{O}\xi(t).$
(66)

Combining (65) with (66), we have that

$\displaystyle\dot{V}(t)+\eta($
$\displaystyle\tau_{j+1}-d)-\eta(\tau_{j}-d)\leq$ (67)
$\displaystyle\xi^{T}(t)\left[\frac{\tau_{j+1}-t}{h}\Upsilon_{1}(h)+\frac{t-\tau_{j}}{h}\Upsilon_{2}(h)\right]\xi(t)$

where

$\displaystyle\Upsilon_{1}(h)$
$\displaystyle=\Xi_{0}+\Psi+\mathcal{O}+h\Xi_{a}+dN\mathcal{T}^{-1}N^{T}$
$\displaystyle\Upsilon_{2}(h)$
$\displaystyle=\Xi_{0}+\Psi+\mathcal{O}+h\Xi_{b}+h\mathcal{M}\mathcal{R}^{-1}\mathcal{M}^{T}+dN\mathcal{T}^{-1}N^{T}.$

Using the Schur Complement Lemma, it can be further deduced that
$\Upsilon_{1}(h)\prec 0$ and $\Upsilon_{2}(h)\prec 0$ are equivalent to (53)
and (57), which are affine in $h$ and $d$. Thus, (53) and (57) at the vertices
of $(h,d)\in[\underline{h},\bar{h}]\times[\underline{d},\bar{d}]$ ensure
$\dot{V}(t)+\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)<0$ for all
$(h,d)\in[\underline{h},\bar{h}]\times[\underline{d},\bar{d}]$. It follows
from [43, Theorem 1] that

$\displaystyle\int_{\tau_{j}}^{\tau_{j+1}}$
$\displaystyle[\dot{V}(s)+\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)]ds=$ (68)
$\displaystyle
V_{a}(\tau_{j+1})-V_{a}(\tau_{j})+h[\eta(\tau_{j+1}-d)-\eta(\tau_{j}-d)]<0$

which implies

$V_{a}(\tau_{j+1})+h\eta(\tau_{j+1}-d)<V_{a}(\tau_{j})+h\eta(\tau_{j}-d),\forall
j\in\mathbb{N}.$ (69)

Finally, using $V_{a}(\tau_{j})>0$ and $\eta(\tau_{j}-d)>0$, we can conclude
that system (3) and $\eta(\tau_{j}-d)$ converge to the origin under our
transmission scheme. This ends the proof. ∎

Theorem 1 is derived based on a novel looped-functional, using which larger
bounds on $h$ can be obtained, compared to the stability condition using only
a common Lyapunov functional. Note that Theorem 1 guarantees stability for
arbitrary values of $h\in[\underline{h},\bar{h}]$ and
$d\in[\underline{d},\bar{d}]$. This is due to the fact that (53) and (57) in
Theorem 1 are affine in $h$ and $d$, and we verify the conditions
simultaneously for $h=\underline{h}$, $h=\bar{h}$, $d=\underline{d}$, and
$d=\bar{d}$.

In networked control systems, the values of the constant sampling interval $h$
and time-delay $d$ may be changed in a set
$[\underline{h},\bar{h}]\times[\underline{d},\bar{d}]$ due to undesired
network effects. Thus, admissible intervals of the time-delay and the sampling
interval are required in co-designing the controller and the triggering scheme
that guarantee the desired system performance. Theorem 1 provides a systematic
approach for computing such regions, based directly on measured data. It is
also an interesting topic to increase the allowable ranges of $h$ and $d$,
which is related to the conservatism of the stability conditions. This issue
is discussed in the following remark.

######  Remark 6 (_Discussion of looped-functional_).

The looped-functional [8] approach has been widely used for stability analysis
of sampled-data control systems. Compared to the continuous-time Lyapunov
stability theorem, the looped-functional approach provides less conservative
stability conditions. Recently, various types of looped-functionals have been
proposed, such as two-sided looped-functional [44], integral-based looped-
functional [11], extended looped-functional [12], and general looped-
functional [45]. In the proof of Theorem 1, a simple type of one-sided delay-
dependent looped-functional $V_{l}(t)$ [cf. (60)] is constructed. It is easy
to prove that
$\lim\limits_{t\rightarrow\tau_{j}}V_{l}(t)=\lim\limits_{t\rightarrow\tau_{j+1}}V_{l}(t)$,
which asserts that $V_{l}(t)$ is a looped-functional for
$t\in[\tau_{j},\tau_{j+1})$. By Theorem 1, upper bounds on $h$ for varying
transmission delays $d$ can be readily computed. The practical applicability
of Theorem 1 is illustrated with numerical examples in Section IV. It should
be mentioned that larger allowable bounds on $h$ can be obtained by using more
complicated looped-functional methods. Besides, we only focus on closed-loop
stability in Theorem 1, but it is straightforward to derive other performance
guarantees, including e.g., on the closed-loop $\mathcal{L}_{2}$-gain, using
similar arguments as in [46].

###  III-B Data-based stability analysis

We now derive a data-based stability certificate for the event-triggered
control system (3) with _unknown_ system matrices $A$ and $B$ (cf. Assumption
1). The main idea is to rebuild a system expression using the data
$\\{\dot{x}(T_{i}),\,x(T_{i}),\,u(T_{i})\\}^{\rho}_{i=1}$ to replace the
matrix $(A~{}B)$-based representation in (1). According to this, a data-based
system representation given in Lemma 1, combined with the model-based
stability condition in Theorem 1, is employed to obtain a data-based stability
condition.

######  Theorem 2 (_Data-based condition with unknown $A$ and $B$_).

For given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\theta\geq 0$, $\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$
satisfying $1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, system (3) is
asymptotically stable under the triggering condition in (47) for any
$[A~{}B]\in\Sigma_{s}$, and $\eta(\tau_{j}-d)$ converges to the origin for any
$\eta(0)\geq 0$, if there exist a scalar $\varepsilon>0$, and matrices $P\succ
0$, $Z\succ 0$, $T\succ 0$, $R_{1}\succ 0$, $R_{2}\succ 0$, $\Omega\succ 0$,
$S=S^{T}$, $N$, $M_{1}$, $M_{2}$, $F$, such that the following LMIs hold for
all $h\in\\{\underline{h},\bar{h}\\}$ and $d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\begin{array}[]{cccc}\mathcal{G}_{3}&\mathcal{G}_{2}+F^{T}&0\\\
\ast&\mathcal{G}_{1}+\Xi_{0}+h\Xi_{a}+\tilde{\Psi}+\mathcal{O}&dN\\\
\ast&\ast&d\mathcal{T}\\\ \end{array}\right]\prec 0$ (73)
$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\mathcal{G}_{3}&\mathcal{G}_{2}+F^{T}&0&0\\\
\ast&\mathcal{G}_{1}+\Xi_{0}+h\Xi_{b}+\tilde{\Psi}+\mathcal{O}&dN&h\mathcal{M}\\\
\ast&\ast&d\mathcal{T}&0\\\
\ast&\ast&\ast&h\mathcal{R}\end{array}\\!\\!\right]\prec 0$ (78)

where

$\displaystyle\tilde{\Psi}$ $\displaystyle:={\rm
Sym}\big{\\{}-FL_{3}\big{\\}}$ $\displaystyle\mathcal{G}_{1}$
$\displaystyle:=\varepsilon\mathcal{Y}_{1}^{T}\tilde{\Theta}\mathcal{Y}_{1},~{}\mathcal{G}_{2}:=\varepsilon\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{1},~{}\mathcal{G}_{3}:=\varepsilon\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{2}$
$\displaystyle\tilde{\Theta}$
$\displaystyle:=\left[\begin{array}[]{cc}-\tilde{R}_{c}&\tilde{S}_{c}^{T}\\\
\ast&-\tilde{Q}_{c}\\\
\end{array}\right],~{}\left[\begin{array}[]{cc}\tilde{Q}_{c}&\tilde{S}_{c}\\\
\ast&\tilde{R}_{c}\\\
\end{array}\right]:=\left[\begin{array}[]{cc}Q_{c}&S_{c}\\\ \ast&R_{c}\\\
\end{array}\right]^{-1}$ $\displaystyle\mathcal{Y}_{1}$
$\displaystyle:=[0~{}L_{1}^{T}~{}(KL_{10})^{T}]^{T},~{}\mathcal{Y}_{2}:=[I~{}0~{}0]^{T}.$

###### Proof.

According to (61) in the proof of Theorem 1, we have a conclusion that the
system (3) under the event-triggering scheme (47) is stable if
$\Upsilon_{1}(h)\prec 0$ and $\Upsilon_{2}(h)\prec 0$. By matrix
decomposition, $\Upsilon_{1}(h)$ and $\Upsilon_{2}(h)$ are rewritten as

$\displaystyle\Upsilon_{1}(h)$
$\displaystyle=\left[\begin{array}[]{cc}Al_{1}+BKl_{19}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{1}(h)+\tilde{\Psi}-\Psi\\\ \end{array}\right][\cdot]$ (83)
$\displaystyle\Upsilon_{2}(h)$
$\displaystyle=\left[\begin{array}[]{cc}Al_{1}+BKl_{19}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{2}(h)+\tilde{\Psi}-\Psi\\\ \end{array}\right][\cdot]$ (88)

Besides, applying the dualization lemma [47, Lemma 4.9] to the system
representation in (19) with Assumption 4, it can be proved that
$[A~{}B]\in\Sigma_{s}$ if and only if

$\displaystyle\left[\\!\\!\begin{array}[]{cc}[A~{}B]\\\ I\\\
\end{array}\\!\\!\right]^{T}\left[\begin{array}[]{cc}-\tilde{R}_{c}&\tilde{S}_{c}^{T}\\\
\ast&-\tilde{Q}_{c}\\\
\end{array}\right]\left[\\!\\!\begin{array}[]{cc}[A~{}B]\\\ I\\\
\end{array}\\!\\!\right]\succeq 0.$ (95)

Using the full-block S-procedure [48], we have $\Upsilon_{1}(h)\prec 0$ and
$\Upsilon_{2}(h)\prec 0$ for any $[A~{}B]\in\Sigma_{s}$ if there exists a
scalar $\varepsilon>0$ such that

$\displaystyle\tilde{\Upsilon}_{1}(h)+\varepsilon\left[\begin{array}[]{cc}\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{2}&\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{1}\\\
\ast&\mathcal{Y}_{1}^{T}\tilde{\Theta}\mathcal{Y}_{1}\\\
\end{array}\right]\prec 0$ (98)
$\displaystyle\tilde{\Upsilon}_{2}(h)+\varepsilon\left[\begin{array}[]{cc}\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{2}&\mathcal{Y}_{2}^{T}\tilde{\Theta}\mathcal{Y}_{1}\\\
\ast&\mathcal{Y}_{1}^{T}\tilde{\Theta}\mathcal{Y}_{1}\\\
\end{array}\right]\prec 0.$ (101)

Then, the Schur Complement Lemma renders (98) and (101) equivalent to (73) and
(78). Similar to Theorem 1, we conclude that system (3) is asymptotically
stable under triggering condition (47) for any $[A~{}B]\in\Sigma_{s}$, and
$\eta(\tau_{j}-d)$ converges to the origin, which completes the proof. ∎

Theorem 2 allows us to analyze stability properties of dynamic event-triggered
control with delays, without any model knowledge (cf. Assumption 1). A
possibly large MSI and a triggering matrix $\Omega$ for the triggering
condition (47) can be searched for with a given controller gain $K$ by using
Theorem 2. The application of Theorem 2 is simple, requiring only the solution
of LMIs which can be constructed based on noisy data. For event-triggered
systems, a larger MSI leads to a smaller transmission frequency, which saves
network transmission resources. To this end, we further investigate a data-
based stability condition for system (3) with _unknown_ state matrix $A$ and
_known_ input matrix $B$ (cf. Assumption 2), since any additional prior
knowledge may lead to a less conservative data-based stability condition
(i.e., a larger MSI) if compared to the one based only on the available data
(cf. Theorem 2). The following stability result is established based on Lemma
2 and Assumption 5.

######  Theorem 3 (_Data-based condition with unknown $A$ and known $B$_).

For given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$ satisfying
$1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, system (3) is
asymptotically stable under the triggering condition in (47) for any
$[A]\in\bar{\Sigma}_{s}$, and $\eta(\tau_{j}-d)$ converges to the origin for
any $\eta(0)\geq 0$, if there exist a scalar $\varepsilon>0$, and matrices
$P\succ 0$, $Z\succ 0$, $R_{1}\succ 0$, $R_{2}\succ 0$, $\Omega\succ 0$,
$S=S^{T}$, $N$, $M_{1}$, $M_{2}$, $F$, such that the following LMIs hold for
all $h\in\\{\underline{h},\bar{h}\\}$ and $d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\hat{\mathcal{G}}_{3}&\hat{\mathcal{G}}_{2}+F^{T}&0\\\
\ast&\hat{\mathcal{G}}_{1}+\Xi_{0}+h\Xi_{a}+\hat{\Psi}+\mathcal{O}&dN\\\
\ast&\ast&d\mathcal{T}\end{array}\\!\\!\right]\\!\\!\prec 0$ (105)
$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\hat{\mathcal{G}}_{3}&\hat{\mathcal{G}}_{2}+F^{T}&\\!0&\\!0\\\
\ast&\hat{\mathcal{G}}_{1}+\Xi_{0}+h\Xi_{b}+\hat{\Psi}+\mathcal{O}&\\!dN&\\!h\mathcal{M}\\\
\ast&\ast&\\!d\mathcal{T}&\\!0\\\
\ast&\ast&\\!\ast&\\!h\mathcal{R}\end{array}\\!\\!\right]\prec 0$ (110)

where

$\displaystyle\hat{\Psi}$ $\displaystyle:={\rm
Sym}\big{\\{}F(BKL_{10}-L_{3})\big{\\}}$ $\displaystyle\hat{\mathcal{G}}_{1}$
$\displaystyle:=\varepsilon\hat{\mathcal{Y}}_{1}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1},~{}\hat{\mathcal{G}}_{2}:=\varepsilon\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1},~{}\hat{\mathcal{G}}_{3}:=\varepsilon\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{2}$
$\displaystyle\hat{\Theta}$
$\displaystyle:=\left[\begin{array}[]{cc}-\hat{R}_{c}&\hat{S}_{c}^{T}\\\
\ast&-\hat{Q}_{c}\\\
\end{array}\right],~{}\left[\begin{array}[]{cc}\hat{Q}_{c}&\hat{S}_{c}\\\
\ast&\hat{R}_{c}\\\
\end{array}\right]:=\left[\begin{array}[]{cc}\bar{Q}_{c}&\bar{S}_{c}\\\
\ast&\bar{R}_{c}\\\ \end{array}\right]^{-1}$
$\displaystyle\hat{\mathcal{Y}}_{1}$
$\displaystyle:=[0~{}L_{1}^{T}]^{T},~{}\hat{\mathcal{Y}}_{2}:=[I~{}0]^{T}.$

###### Proof.

The full proof is similar to that of Theorem 2. Terms $\Upsilon_{1}(h)$ and
$\Upsilon_{2}(h)$ in (83) and (88) can be rebuilt as follows

$\displaystyle\Upsilon_{1}(h)$
$\displaystyle=\left[\begin{array}[]{cc}Al_{1}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{1}(h)+\hat{\Psi}-\Psi\\\
\end{array}\right]\left[\begin{array}[]{cc}Al_{1}\\\ I\\\ \end{array}\right]$
(117) $\displaystyle\Upsilon_{2}(h)$
$\displaystyle=\left[\begin{array}[]{cc}Al_{1}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{2}(h)+\hat{\Psi}-\Psi\\\
\end{array}\right]\left[\begin{array}[]{cc}Al_{1}\\\ I\\\ \end{array}\right]$
(124)

Employing the dualization lemma to the system representation in (34) and
Assumption 5, we have that $[A]\in\bar{\Sigma}_{s}$ if and only if

$\displaystyle\left[\begin{array}[]{cc}A\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}-\hat{R}_{c}&\hat{S}_{c}^{T}\\\
\ast&-\hat{Q}_{c}\\\ \end{array}\right]\left[\begin{array}[]{cc}A\\\ I\\\
\end{array}\right]\succeq 0.$ (131)

According to the full-block S-procedure, it holds that $\Upsilon_{1}(h)\prec
0$ and $\Upsilon_{2}(h)\prec 0$ for any $[A]\in\bar{\Sigma}_{s}$ if there
exists a scalar $\varepsilon>0$ such that

$\displaystyle\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{1}(h)+\hat{\Psi}-\Psi\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{2}&\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1}\\\
\ast&\hat{\mathcal{Y}}_{1}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1}\\\
\end{array}\right]\prec 0$ (136)
$\displaystyle\left[\begin{array}[]{cc}0&F^{T}\\\
\ast&\Upsilon_{2}(h)+\hat{\Psi}-\Psi\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{2}&\hat{\mathcal{Y}}_{2}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1}\\\
\ast&\hat{\mathcal{Y}}_{1}^{T}\hat{\Theta}\hat{\mathcal{Y}}_{1}\\\
\end{array}\right]\prec 0.$ (141)

Finally, similar to Theorem 1, we conclude that (105) and (110) are sufficient
stability conditions for system (3) under the triggering condition (47) for
any $[A]\in\bar{\Sigma}_{s}$, and $\eta(\tau_{j}-d)$ converges to the origin.
∎

Theorem 3 provides a stability condition of system (1) using the prior
knowledge of the input matrix $B$ and data. Compared to Theorem 2 based only
on the available data, a larger MSI is able to be obtained by Theorem 3 for a
given controller gain; see numerical comparisons between Theorems 2 and 3 for
different levels of noise given in Table I of Section IV.

###  III-C Data-based co-designing controller and triggering matrix

This section provides co-design methods of the controller gain $K$ and the
triggering matrix $\Omega$ for system (3) under Assumptions 1 and 2. For
Assumptions 1, we want to compute matrices $K$ and $\Omega$ using Theorem 2
with a given sampling interval $h$ and the delay $d$. However, since the
matrix $\tilde{Q}_{c}$ is generally not negative definite, inequalities (73)
and (78) are actually not convex in $K$. It is hence hard to design $K$
directly through Theorem 2. Inspired by the iterative approach in [32,
Theorems 4-5], we give a reformulation of Theorem 2 that is convex in $K$ for
a fixed variable $F$. Therefore, the controller gain can be searched while
simultaneously obtaining an MSI under Assumption 1.

######  Theorem 4 (_Co-designing under unknown $A$, $B$, and fixed matrix
$F$_).

For any given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$ satisfying
$1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, there exists a controller
gain $K$ such that system (3) is asymptotically stable under the triggering
condition in (47) for any pair $[A~{}B]\in\Sigma_{s}$, and that
$\eta(\tau_{j}-d)$ converges to the origin for any $\eta(0)\geq 0$, provided
that there are $\varepsilon>0$, and matrices $P\succ 0$, $Z\succ 0$, $T\succ
0$, $R_{1}\succ 0$, $R_{2}\succ 0$, $\Omega\succ 0$, $S=S^{T}$, $N$, $M_{1}$,
$M_{2}$, $F$ such that the following LMIs hold for all
$h\in\\{\underline{h},\bar{h}\\}$ and $d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\mathcal{S}_{1}&\mathcal{S}_{2}+\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}&0\\\
\ast&\mathcal{S}_{3}+\Xi_{0}+h\Xi_{a}+\tilde{\Psi}+\mathcal{O}&dN\\\
\ast&\ast&d\mathcal{T}\\\ \ast&\ast&\ast\end{array}\\!\\!\right]\\!\\!\prec 0$
(146)
$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\mathcal{S}_{1}&\mathcal{S}_{2}+\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}&0&0\\\
\ast&\mathcal{S}_{3}+\Xi_{0}+h\Xi_{b}+\tilde{\Psi}+\mathcal{O}&dN&h\mathcal{M}\\\
\ast&\ast&d\mathcal{T}&0\\\
\ast&\ast&\ast&h\mathcal{R}\end{array}\\!\\!\right]\prec 0$ (151)

where

$\displaystyle\mathcal{S}_{1}$
$\displaystyle:=\varepsilon\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{1}^{T},\mathcal{S}_{2}:=\varepsilon\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{2}^{T},\mathcal{S}_{3}:=\varepsilon\mathcal{D}_{2}\Theta_{s}\mathcal{D}_{2}^{T}$
$\displaystyle\mathcal{D}_{1}$
$\displaystyle:=\left[\begin{array}[]{ccc}I&0&0\\\ 0&1&0\\\
\end{array}\right],~{}\mathcal{D}_{2}:=\left[\begin{array}[]{ccc}0&0&F^{T}\\\
\end{array}\right].$

###### Proof.

Similar to the proof of Theorem 2, we reformulate $\Upsilon_{1}(h)$ and
$\Upsilon_{2}(h)$ as follows

$\displaystyle{\Upsilon}_{1}(h)=\left[\begin{array}[]{cc}[FA~{}FB]^{T}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}\\\
\ast&\Upsilon_{1}(h)+\tilde{\Psi}-\Psi\\\ \end{array}\right][\cdot]$ (156)
$\displaystyle{\Upsilon}_{2}(h)=\left[\begin{array}[]{cc}[FA~{}FB]^{T}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}\\\
\ast&\Upsilon_{2}(h)+\tilde{\Psi}-\Psi\\\ \end{array}\right][\cdot].$ (161)

From the data-based representation (19), it holds that

$\displaystyle\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\ I\\\
\end{array}\\!\right]^{T}\\!\Theta_{s}\\!\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\
I\\\ \end{array}\\!\right]\succeq 0.$ (166)

By the full-block S-procedure, we have ${\Upsilon}_{1}(h)\prec 0$ and
${\Upsilon}_{2}(h)\prec 0$ for any $[A~{}B]\in\Sigma_{s}$ if there exists a
scalar $\varepsilon>0$ such that

$\displaystyle\left[\begin{array}[]{cc}0&\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}\\\
\ast&\Upsilon_{1}(h)+\tilde{\Psi}-\Psi\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{1}^{T}&\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{2}^{T}\\\
\ast&\mathcal{D}_{2}\Theta_{s}\mathcal{D}_{2}^{T}\\\ \end{array}\right]$ (171)
$\displaystyle\prec 0$ (172)
$\displaystyle\left[\begin{array}[]{cc}0&\big{[}L_{1}^{T}~{}(KL_{10})^{T}\big{]}^{T}\\\
\ast&\Upsilon_{2}(h)+\tilde{\Psi}-\Psi\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{1}^{T}&\mathcal{D}_{1}\Theta_{s}\mathcal{D}_{2}^{T}\\\
\ast&\mathcal{D}_{2}\Theta_{s}\mathcal{D}_{2}^{T}\\\ \end{array}\right]$ (177)
$\displaystyle\prec 0.$ (178)

Finally, similar to the proof of Theorem 2, we have a conclusion that (146)
and (151) are sufficient stability conditions for system (3) under the
triggering condition (47) for any $[A~{}B]\in\Sigma_{s}$, and
$\eta(\tau_{j}-d)$ converges to the origin, which completes the proof. ∎

######  Remark 7 (_Iterative approach_).

Notice that the inequalities (146) and (151) in Theorem 4 are quadratic in the
matrix $F$. Hence, we cannot efficiently search for the matrix $K$ by Theorem
4. Fixing the matrix $F$ enables the design of the controller; however, the
conservatism of the obtained MSI estimate may be significantly increased. To
this end, we propose an iterative approach which alternates between solving
the LMIs in Theorems 2 and 4 while fixing the matrix $K$ or $F$, respectively.
The effectiveness of this method is illustrated via a numerical example in
Section IV, and the results of MSI bounds in Table II show that such an
iterative method tremendously improves the MSI bounds in comparison to using a
fixed controller gain as in Theorems 2 and 3 (see Table I). On the other hand,
for the case of having a known input matrix $B$, we can search for a
controller gain $K$ with possibly large MSI by alternating between solving the
LMIs in Theorem 3 while fixing matrix $K$ or matrix $F$, in both of which the
LMIs are convex and can be efficiently solved. Compared with the iterative
method between Theorems 2 and 4 that are purely based on data, the one using
Theorem 3 that contains the prior knowledge of matrix $B$ may provide a larger
MSI while iteratively searching for the corresponding $K$; see numerical
comparisons in Table II.

In the following, we provide a stability condition that is convex in all
decision variables (different from Theorems 3, 4 and [32, Theorem 5]) for co-
designing matrices $K$ and $\Omega$, at the price of additional conservatism
with respect to the matrix $F$. For this purpose, we begin with an
algebraically equivalent transformation of system (3).

Let $G\in\mathbb{R}^{n\times n}$ be a nonsingular matrix, and define
$x(t)=Gz(t)$. system (3) is transformed into

$\displaystyle\dot{z}(t)=G^{-1}AGz(t)+G^{-1}BK_{c}z(t_{k}-d),~{}~{}~{}~{}t\in[t_{k},t_{k+1})$
(179)

where $K_{c}:=KG$.

System (179) exhibits the same stability behavior as system (3), and the
triggering condition (47) is still valid. Based on Theorem 1 and the
equivalent system expression in (179), we have the following stability result
under Assumption 1.

######  Theorem 5 (_Co-designing under unknown $A$ and $B$_).

For given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$ satisfying
$1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, there exists a controller
gain $K$ such that system (3) is asymptotically stable under the triggering
condition in (47) for any $[A~{}B]\in\Sigma_{s}$, and $\eta(\tau_{j}-d)$
converges to the origin for any $\eta(0)\geq 0$, if there exist scalars
$\varepsilon>0$, $\epsilon>0$, and matrices $P\succ 0$, $Z\succ 0$, $T\succ
0$, $R_{1}\succ 0$, $R_{2}\succ 0$, $\Omega\succ 0$, $S=S^{T}$, $N$, $M_{1}$,
$M_{2}$, $G$, $K_{c}$ such that the following LMIs hold for all
$h\in\\{\underline{h},\bar{h}\\}$ and $d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\mathcal{B}_{1}&\mathcal{B}_{2}+\big{[}GL_{1}^{T}~{}K_{c}L_{10}^{T}\big{]}^{T}&0\\\
\ast&\mathcal{B}_{3}+\Xi_{0}+h\Xi_{a}+\check{\Psi}+\mathcal{O}&dN\\\
\ast&\ast&d\mathcal{T}\\\ \ast&\ast&\ast\end{array}\\!\\!\right]\\!\\!\prec 0$
(184)
$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\mathcal{B}_{1}&\mathcal{B}_{2}+\big{[}GL_{1}^{T}~{}K_{c}L_{10}^{T}\big{]}^{T}&0&0\\\
\ast&\mathcal{B}_{3}+\Xi_{0}+h\Xi_{b}+\check{\Psi}+\mathcal{O}&dN&h\mathcal{M}\\\
\ast&\ast&d\mathcal{T}&0\\\
\ast&\ast&\ast&h\mathcal{R}\end{array}\\!\\!\right]\prec 0$ (189)

where

$\displaystyle\check{\Psi}$ $\displaystyle:={\rm
Sym}\big{\\{}-(L_{1}^{T}+\epsilon L_{3}^{T})GL_{3}\big{\\}}$
$\displaystyle\mathcal{B}_{1}$
$\displaystyle:=\varepsilon\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{1}^{T},~{}\mathcal{B}_{2}:=\varepsilon\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{2}^{T},~{}\mathcal{B}_{3}:=\varepsilon\mathcal{V}_{2}\Theta_{s}\mathcal{V}_{2}^{T}$
$\displaystyle\mathcal{V}_{1}$
$\displaystyle:=\left[\begin{array}[]{ccc}I&0&0\\\ 0&1&0\\\
\end{array}\right],~{}\mathcal{V}_{2}:=\left[\begin{array}[]{ccc}0&0&(L_{1}^{T}+\epsilon
L_{3}^{T})\\\ \end{array}\right].$

Moreover, the controller gain $K$ is given by $K=K_{c}G^{-1}$.

###### Proof.

We construct a functional $V(z,t)$ for system (179) by substituting $x$ of the
functional $V(x,t)$ in (58) with $z$. Similar to the proof of Theorem 1, it
can be obtained that

$\displaystyle\dot{V}(z,t)\leq$
$\displaystyle~{}\xi^{T}(z,t)\left[\frac{\tau_{j+1}-t}{h}\Upsilon_{1}(h)+\frac{t-\tau_{j}}{h}\Upsilon_{2}(h)\right]\xi(z,t).$
(190)

According to system (179), we have that

$\displaystyle{}0=$ $\displaystyle
2\big{[}z(t)+\epsilon\dot{z}(t)\big{]}\times$ $\displaystyle\times
G\big{[}G^{-1}AGz(t)+G^{-1}BK_{c}z(t_{k}-d)-\dot{z}(t)\big{]}.$ (191)

Replacing the equality (III-A) by (III-C) in the proof of Theorem 1, it can be
obtained that

$\displaystyle\dot{V}(z,t)\leq$
$\displaystyle~{}\xi^{T}(z,t)\left[\frac{\tau_{j+1}-t}{h}\check{\Upsilon}_{1}(h)+\frac{t-\tau_{j}}{h}\check{\Upsilon}_{2}(h)\right]\xi(z,t).$
(192)

where $\check{\Upsilon}_{i}(h):={\rm Sym}\big{\\{}(L_{1}^{T}+\epsilon
L_{3}^{T})(AGL_{1}+BK_{c}L_{10})\big{\\}}+{\Upsilon}_{i}(h)-{\Psi}+\check{\Psi}$
for $i=1,2$. Then, terms $\check{\Upsilon}_{1}(h)$ and
$\check{\Upsilon}_{2}(h)$ are rewritten as

$\displaystyle\check{\Upsilon}_{i}(h)=\left[\\!\begin{array}[]{cc}(L_{1}^{T}+\epsilon
L_{3}^{T})[A~{}B]^{T}\\\ I\\\
\end{array}\\!\right]^{T}\left[\\!\begin{array}[]{cc}0\\!&\\!\big{[}GL_{1}^{T}~{}K_{c}L_{10}^{T}\big{]}^{T}\\\
\ast\\!&\\!\Upsilon_{i}(h)-\Psi+\check{\Psi}\\\ \end{array}\\!\right][\cdot]$
(197)

According to the data-based representation in (19), it holds that

$\displaystyle\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\ I\\\
\end{array}\\!\right]^{T}\\!\Theta_{s}\\!\left[\\!\begin{array}[]{cc}[A~{}B]^{T}\\\
I\\\ \end{array}\\!\right]\succeq 0.$ (202)

By the full-block S-procedure, we have $\check{\Upsilon}_{1}(h)\prec 0$ and
$\check{\Upsilon}_{2}(h)\prec 0$ for any $[A~{}B]\in\Sigma_{s}$ if there
exists a scalar $\varepsilon>0$ such that

$\displaystyle\left[\begin{array}[]{cc}0&\big{[}GL_{1}^{T}~{}K_{c}L_{10}^{T}\big{]}^{T}\\\
\ast&\Upsilon_{1}(h)-\Psi+\check{\Psi}\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{1}^{T}&\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{2}^{T}\\\
\ast&\mathcal{V}_{2}\Theta_{s}\mathcal{V}_{2}^{T}\\\ \end{array}\right]\prec
0$ (207)
$\displaystyle\left[\begin{array}[]{cc}0&\big{[}GL_{1}^{T}~{}K_{c}L_{10}^{T}\big{]}^{T}\\\
\ast&\Upsilon_{2}(h)-\Psi+\check{\Psi}\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{1}^{T}&\mathcal{V}_{1}\Theta_{s}\mathcal{V}_{2}^{T}\\\
\ast&\mathcal{V}_{2}\Theta_{s}\mathcal{V}_{2}^{T}\\\ \end{array}\right]\prec
0.$ (212)

Finally, similar to the proof of Theorem 2, we have a conclusion that (184)
and (189) are sufficient stability conditions for System (179) under the
triggering condition (47) for any $[A~{}B]\in\Sigma_{s}$, and
$\eta(\tau_{j}-d)$ converges to the origin. Since $G$ is a nonsingular matrix,
system (179) exhibits the same stability behavior as system (3), which
completes the proof. ∎

######  Remark 8 (_Novelty_).

The design procedure of Theorem 5 requires no model knowledge, similar to
[32], but only a single open-loop data trajectory affected by noise. Note that
LMIs (184) and (189) are convex in all decision variables, in contrast to
Theorems 2-4 and [32, Theorems 4-5]. We can obtain a controller gain $K$ with
an allowable MSI directly by solving the LMIs in Theorem 5 without fixing any
variables. In essence, Theorem 5 is established at the price of increasing the
conservatism of free matrix $F$; that is, replacing the equality (III-A) with
(III-C) in the proof of Theorem 1. As a result, the free matrix $F$ of
dimensions $10n\times n$ is limited to the matrix $G$ of $n\times n$. That is
to say, Theorem 5 is more conservative than Theorems 2-4. This conclusion is
further corroborated through a numerical comparison in Table III. Besides, in
contrast to Theorems 2–3 and [32, Theorem 4], the restriction of requiring an
invertible matrix $\Theta_{s}$ (cf. Assumption 4) is removed in Theorem 5. The
main reason is that the terms in (197) match the inequality (19) of Lemma 1
exactly. We then do not need the intermediate step of computing
$\Theta_{s}^{-1}$ (namely, (95) in Theorem 2) when designing the controller
gain $K$ by means of Theorem 5. Thus, this incurs lower computational
complexity and no restriction (Assumption 4) in co-designing the controller
gain and the triggering matrix $\Omega$.

Next, we consider using the prior knowledge of input matrix $B$ for obtaining
a larger MSI of system (3) under the triggering condition (47) in the co-
design method of Theorem 5. For this purpose, a co-design method of the
controller gain $K$ and the triggering matrix $\Omega$ for system (3) with an
_unknown_ state matrix $A$ and any _known_ input matrix $B$ (cf. Assumption 2)
is given based on Theorem 5.

######  Theorem 6 (_Co-designing under unknown $A$ and known $B$_).

For given scalars $\bar{h}>\underline{h}>0$, $\bar{d}>\underline{d}>0$,
$\sigma_{1}>0$, $\sigma_{2}>0$, $\lambda>0$, and $\theta>0$ satisfying
$1-\lambda-\frac{1}{\theta}\geq 0$, or $\theta=0$, there exists a controller
gain $K$ such that system (3) is asymptotically stable under the triggering
condition in (47) for any $[A]\in\bar{\Sigma}_{s}$, and $\eta(\tau_{j}-d)$
converges to the origin for any $\eta(0)\geq 0$, if there exist scalars
$\varepsilon>0$, $\epsilon>0$, and matrices $P\succ 0$, $Z\succ 0$, $T\succ
0$, $R_{1}\succ 0$, $R_{2}\succ 0$, $\Omega\succ 0$, $S=S^{T}$, $N$, $M_{1}$,
$M_{2}$, $G$, $K_{c}$ such that the following LMIs hold for
$h\in\\{\underline{h},\bar{h}\\}$ and $d\in\\{\underline{d},\bar{d}\\}$

$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\grave{\mathcal{B}}_{1}&\grave{\mathcal{B}}_{2}+Gl_{1}&0\\\
\ast&\grave{\mathcal{B}}_{3}+\Xi_{0}+h\Xi_{a}+\grave{\Psi}+\mathcal{O}&dN\\\
\ast&\ast&d\mathcal{T}\\\ \ast&\ast&\ast\end{array}\\!\\!\right]\\!\\!\prec 0$
(217)
$\displaystyle\left[\\!\\!\begin{array}[]{cccc}\grave{\mathcal{B}}_{1}&\grave{\mathcal{B}}_{2}+Gl_{1}&0&0\\\
\ast&\grave{\mathcal{B}}_{3}+\Xi_{0}+h\Xi_{b}+\grave{\Psi}+\mathcal{O}&dN&h\mathcal{M}\\\
\ast&\ast&d\mathcal{T}&0\\\
\ast&\ast&\ast&h\mathcal{R}\end{array}\\!\\!\right]\prec 0$ (222)

where

$\displaystyle\grave{\Psi}$ $\displaystyle:={\rm
Sym}\big{\\{}(L_{1}^{T}+\epsilon L_{3}^{T})(BK_{c}l_{19}-Gl_{18})\big{\\}}$
$\displaystyle\grave{\mathcal{B}}_{1}$
$\displaystyle:=\varepsilon\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{1}^{T},~{}\grave{\mathcal{B}}_{2}:=\varepsilon\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{2}^{T},~{}\grave{\mathcal{B}}_{3}:=\varepsilon\grave{\mathcal{V}}_{2}\Theta_{s}\grave{\mathcal{V}}_{2}^{T}$
$\displaystyle\grave{\mathcal{V}}_{1}$
$\displaystyle:=\left[\begin{array}[]{ccc}I&0\\\
\end{array}\right],~{}\grave{\mathcal{V}}_{2}:=\left[\begin{array}[]{ccc}0&(L_{1}^{T}+\epsilon
L_{3}^{T})\\\ \end{array}\right].$

Moreover, the controller gain $K$ is given by $K=K_{c}G^{-1}$.

###### Proof.

The whole proof is similar to Theorem 5, except that we re-express terms
$\check{\Upsilon}_{1}(h)$ and $\check{\Upsilon}_{2}(h)$ as, for $i=1,2$

$\displaystyle\check{\Upsilon}_{i}(h)=\left[\begin{array}[]{cc}[(L_{1}^{T}+\epsilon
L_{3}^{T})A]^{T}\\\ I\\\
\end{array}\right]^{T}\left[\begin{array}[]{cc}0&Gl_{1}\\\
\ast&\Upsilon_{i}(h)-\Psi+\grave{\Psi}\\\ \end{array}\right][\cdot]$ (227)

Then, from the data-based representation (34), it holds that

$\displaystyle\left[\\!\begin{array}[]{cc}A^{T}\\\ I\\\
\end{array}\\!\right]^{T}\\!\Theta_{s}\\!\left[\\!\begin{array}[]{cc}A^{T}\\\
I\\\ \end{array}\\!\right]\succeq 0.$ (232)

Through the full-block S-procedure, it holds that
$\check{\Upsilon}_{1}(h)\prec 0$ and $\check{\Upsilon}_{2}(h)\prec 0$ for any
$[A]\in\bar{\Sigma}_{s}$ if there exists a scalar $\varepsilon>0$ such that

$\displaystyle\left[\begin{array}[]{cc}0&Gl_{1}\\\
\ast&\Upsilon_{1}(h)-\Psi+\grave{\Psi}\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{1}^{T}&\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{2}^{T}\\\
\ast&\grave{\mathcal{V}}_{2}\Theta_{s}\grave{\mathcal{V}}_{2}^{T}\\\
\end{array}\right]\prec 0$ (237)
$\displaystyle\left[\begin{array}[]{cc}0&Gl_{1}\\\
\ast&\Upsilon_{2}(h)-\Psi+\grave{\Psi}\\\
\end{array}\right]+\varepsilon\left[\begin{array}[]{cc}\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{1}^{T}&\grave{\mathcal{V}}_{1}\Theta_{s}\grave{\mathcal{V}}_{2}^{T}\\\
\ast&\grave{\mathcal{V}}_{2}\Theta_{s}\grave{\mathcal{V}}_{2}^{T}\\\
\end{array}\right]\prec 0.$ (242)

Finally, similar to Theorem 5, we conclude that (217) and (222) are sufficient
stability conditions for system (3) under the triggering condition (47) for
any $[A]\in\bar{\Sigma}_{s}$, and $\eta(\tau_{j}-d)$ converges to the origin.
∎

Theorem 6 provides a co-design method for obtaining matrices $K$ and $\Omega$.
Compared to Theorem 5, which is based only on the data, a larger MSI bounds
may be computed by Theorem 6; see the numerical comparison between Theorems 5
and 6 for different noise bounds given in Table III.

######  Remark 9 (_Alternative approach_).

Sampled-data control systems, in which information is transmitted over wired
or wireless networks having limited bandwidth [49], have been widely studied
over the last two decades. Event-triggered control has emerged as a paradigm
for sampled-data systems [50]. In existing works on event-triggered control,
exact knowledge of a plant is assumed. However, in practical situations, it is
often difficult if not impossible to acquire an accurate system model. This
motivates well the combination of data-driven control and event-triggering
control for sampled-data systems. In our proposed approach, the controller and
the triggering protocol are simultaneously designed without explicit knowledge
of the system model. To the authors’ knowledge, the only alternative to our
approach in the current literature would be first performing system
identification, using e.g., least-squares estimation of the system matrices,
followed by event-triggered control that has been investigated in [35, 37].
However, such an identification-based event-triggered control does in general
not provide stability guarantees, in particular when the data are affected by
noise. This is due to the fact that i) providing tight estimation bounds for
system identification is challenging, compare e.g., [18]; and ii) [35, 37]
only provide nominal results, i.e., potential error bounds, e.g., arising from
system identification are not handled systematically. In contrast, our event-
triggered control approach admits rigorous stability guarantees based directly
on noisy data.

##  IV Example and Simulation

Two numerical examples are presented in this section to demonstrate the
effectiveness of our proposed methods.

######  Example 1.

_Consider system ( 3) under the transmission scheme (47) with parameters in
[49]_

$A=\left[\begin{array}[]{cc}0&1\\\ 0&-0.1\\\
\end{array}\right],~{}~{}B=\left[\begin{array}[]{cc}0\\\ 0.1\\\
\end{array}\right].$

_Assume that the matrices $A$ and $B$ are _unknown_. We consider data-driven
event-triggered control of system (3). To this end, we generate measurements
$\\{\dot{x}(T_{i}),\,x(T_{i}),\,u(T_{i})\\}^{100}_{i=1}$ with the interval
$T_{i+1}-T_{i}$ satisfying_

$T_{i+1}-T_{i}=\left\\{\begin{array}[]{lll}1&,&t\in[1,\,49]\\\
2&,&t\in[50,\,99]\end{array}\right.$ (243)

_where the data-generating input was sampled uniformly from
$u(t)\in[-1,~{}1]$. The measured data were perturbed by a disturbance
distributed uniformly over $w(t)\in[-\bar{w},\,\bar{w}]$ for $\bar{w}>0$.
According to Remark 1, $w(t)$ satisfies Assumption 3 with $Q_{d}=-I$,
$S_{d}=0$, and $R_{d}=\bar{w}^{2}\rho I$ $(\rho=100)$._

TABLE I: Maximum allowable $\bar{h}$ with $\underline{h}=10^{-5}$ and delay $\underline{d}=0$ for different bounds $\bar{w}$, $\bar{d}$, and given $K=-[3.75~{}11.5]$. $\bar{w}$ | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05
---|---|---|---|---|---|---
Theorem 2 ($\bar{d}=0$) | 1.04 | 0.96 | 0.81 | 0.72 | 0.64 | 0.57
Theorem 3 ($\bar{d}=0$) | 1.12 | 1.09 | 1.04 | 1.00 | 0.99 | 0.91
Theorem 2 ($\bar{d}=0.1$) | 0.78 | 0.70 | 0.58 | 0.47 | 0.38 | 0.31
Theorem 3 ($\bar{d}=0.1$) | 0.86 | 0.84 | 0.82 | 0.76 | 0.73 | 0.71
Theorem 2 ($\bar{d}=0.2$) | 0.65 | 0.57 | 0.44 | 0.35 | 0.27 | 0.21
Theorem 3 ($\bar{d}=0.2$) | 0.73 | 0.71 | 0.68 | 0.64 | 0.61 | 0.59

(_MSI bounds_) Using Theorem 2 and controller gain $K=-[3.75~{}11.5]$,
possibly large values of $\bar{h}$ with $\underline{h}=10^{-5}$ and
$\underline{d}=0$ for different realizations of $\bar{w}$ and $\bar{d}$ were
computed and are presented in Table I. It can be observed that larger values
of $\bar{w}$ lead to a higher sampling frequency to guarantee stability, which
is consistent with [32]. Besides, from Table I, the increase of transmission
delay reduces the MSI bound, i.e., more transmissions are required for
stability.  Furthermore, we computed possibly large bounds $\bar{h}$ for
different $\bar{w}$ by using Theorem 3, i.e., assuming that the input matrix B
is known. As shown in Table I, Theorem 3 provides larger values of $\bar{h}$
than Theorem 2, which confirms that additional prior knowledge may lead to
less conservative data-based stability conditions compared with those
established solely on the available data.

In Table II, values of $\bar{h}$ associated with the controller gain $K$
iteratively found by [32, Theorems 4-5], Theorems 2 and 4, as well as Theorem
3 are displayed. There is a similar conclusion as in Table I that the
iterative approach using Theorem 3 enhances robustness to noise relative to
[32, Theorems 4-5] and Theorem 2 with Theorem 4. Except for this, by choosing
a different controller gain, improved MSI bounds can be obtained by the
iterative approaches in Table II when compared to the ones in Table I obtained
by the fixed controller gain. Table III depicts values of $\bar{h}$ found by
Theorems 5-6 with $\epsilon=2$. Theorems 5 and 6 provide smaller $h$ values
than Theorems 2-3, which confirms usefulness of the free matrix $F$ in
reducing stability conservatism. It is worth noting that, at the price of
additional conservatism with respect to matrix $F$, Theorems 5 and 6 offer
convex LMI conditions. Different from Theorems 2-4 and [32, Theorems 4-5],
matrices $K$ and $\Omega$ can be co-designed directly based on Theorems 5 and
6.

TABLE II: Maximum allowable $\bar{h}$ with $\underline{h}=10^{-5}$ and delay $d=0$ for different noise bounds $\bar{w}$ by iterative approaches. $\bar{w}$ | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05
---|---|---|---|---|---|---
Theorems 2 and 4 | 23.4 | 14.7 | 5.2 | 3.4 | 2.1 | 1.9
[32, Theorems 4-5] | 28.5 | 13.8 | 6.3 | 4.0 | 2.9 | 2.2
Theorem 3 | 32.2 | 18.8 | 8.1 | 4.8 | 3.5 | 2.6
TABLE III: Maximum allowable $\bar{h}$ with $\underline{h}=10^{-5}$ and delay $d=0$ for different noise bounds $\bar{w}$ and the optimized controller gain $K$. $\bar{w}$ | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05
---|---|---|---|---|---|---
Theorem 5 | 11.6 | 8.4 | 4.2 | 2.7 | 1.9 | 1.6
Theorem 6 | 14.0 | 10.9 | 5.1 | 3.2 | 2.0 | 1.8

(_Co-designing under unknown $A$ and $B$_) For data-driven control of system
(3) with unknown matrices $A$ and $B$ under our dynamic transmission scheme
(47), we employ Theorem 5 to co-design the controller gain $K$ and triggering
matrix $\Omega$. We consider constant transmission delay bounds
$\underline{d}=0$, $\bar{d}=0.1$, sampling interval $h=0.2$, triggering
parameters $\sigma_{1}=0.4$, $\sigma_{2}=0.1$, $\theta=2$, $\lambda=0.5$,
$\epsilon=2$, noise $w(t)\in[-0.01,~{}0.01]$, and the available measurements
$\\{\dot{x}(T_{i}),~{}x(T_{i}),~{}u(T_{i})\\}^{100}_{i=1}$ with the interval
$T_{i+1}-T_{i}$ satisfying (243). Solving the data-based LMIs (184) and (189),
the controller gain and the triggering matrix were co-designed as follows

$\displaystyle\Omega$
$\displaystyle=\left[\begin{array}[]{cccc}218.2530&-59.7385\\\
-59.7385&95.9565\\\ \end{array}\right]$ $\displaystyle K$
$\displaystyle=\left[\begin{array}[]{cccc}-0.1798&-3.7008\end{array}\right].$

We then simulated system (3) under the triggering scheme in (47) with initial
condition $x(0)=[3,\,-2]^{T}$, as well as the dynamic variable
$\eta(\tau_{j}-d)$, over the time interval $[0,30]$. Their trajectories are
depicted in Fig. 3. Evidently, both the system states and the dynamic variable
converge to the origin, which demonstrates the feasibility of our designed
controller gain $K$ and the triggering matrix $\Omega$. Additionally, Fig. 3
plots the evolutions of error function $\iota f_{c}$ and threshold $\iota
f_{t}$, where $\iota:=\tau_{j}^{3}$ is a weighting function for visualizing
triggers, and

$\displaystyle f_{c}$ $\displaystyle:=e^{T}(\tau_{j}-d)\Omega e(\tau_{j}-d),$
$\displaystyle f_{t}$
$\displaystyle:=\frac{1}{\theta}\eta(\tau_{j}-d)+\sigma_{1}x^{T}(\tau_{j}-d)\Omega
x(\tau_{j}-d)$ $\displaystyle~{}~{}~{}~{}+\sigma_{2}x^{T}(t_{k}-d)\Omega
x(t_{k}-d).$

It is evident from the plots that an event is triggered as soon as the
triggering condition $\iota f_{c}$ exceeds the threshold $\iota f_{t}$. At
that triggering instant, the state $x(\tau_{j}-d)$ is transmitted to the
controller, and the measurement error $e(\tau_{j}-d)$ is set to be zero in
$\eta(\tau_{j}-d)$. Thus, consistent with the theoretical result in (48), the
error $\iota f_{c}$ is kept smaller than the threshold $\iota f_{t}$.  It is
also worth pointing out that only $20$ measurements were transmitted to the
controller under our proposed triggering scheme (47), while $150$ measurements
were sampled. This validates the effectiveness of the proposed scheme in
saving communication resources, while maintaining system stability.

Figure 3: Trajectories of system (3), dynamic variable $\eta(\tau_{j}-d)$,
error function $f_{c}$, and threshold $f_{t}$ under triggering condition (47)
with parameters $h=0.2$, $\sigma_{1}=0.4$, $\sigma_{2}=0.1$, $\theta=1$,
$\lambda=0.5$, unknown matrices $A$ and $B$ with initial condition
$x(0)=[3,\,-2]^{T}$, and delay $d=0.1$.

Figure 4: Trajectories of system (3), dynamic variable $\eta(\tau_{j}-d)$,
error function $f_{c}$, and threshold $f_{t}$ under triggering condition (47)
with parameters $h=0.2$, $\sigma_{1}=0.4$, $\sigma_{2}=0.1$, $\theta=2$,
$\lambda=0.5$, unknown matrix $A$ and known $B$ with initial condition
$x(0)=[3,\,-2]^{T}$, and delay $d=0.1$.

(_Co-designing under unknown $A$ and known $B$_) For the event-triggered
system (3) with unknown $A$ and known $B$ under the dynamic transmission
scheme (47), we considered the same parameters as used in Fig. 3 and the
available measurements
$\\{\dot{x}(T_{i}),~{}x(T_{i}),~{}u(T_{i})\\}^{100}_{i=1}$ with the interval
$T_{i+1}-T_{i}$ satisfying (243). Solving the LMIs (217) and (222), the co-
designed controller gain and triggering matrix leveraging Theorem 6 were
computed as follows

$\displaystyle\Omega$
$\displaystyle=\left[\begin{array}[]{cccc}135.0611&-65.0218\\\
-65.0218&292.8338\\\ \end{array}\right]$ $\displaystyle K$
$\displaystyle=\left[\begin{array}[]{cccc}-0.4101&-8.0184\end{array}\right].$

The simulation results are shown in Fig. 4. Both the system states and the
dynamic variable converged to the origin, while 15 measurements out of 150
sampled-data were transmitted to the controller. Clearly, the co-design method
in Theorem 6 is effective for System (3) with unknown system matrix $A$ and
known matrix $B$. Last but not least, by comparing Figs. 3 and 4, the
knowledge of $B$ allows to stabilize the system using fewer transmissions.

Figure 5: Trajectories of system (3), dynamic variable $\eta(\tau_{j})$ under
triggering condition (47) with parameters $h=0.1$, $\sigma_{1}=0.01$,
$\sigma_{2}=0.01$, $\theta=2$, $\lambda=0.5$, unknown matrices $A$ and $B$
with initial condition $x(0)=[0.98,~{}0,~{}0.2,~{}0]^{T}$, and delay $d=0.01$.

######  Example 2.

_The second example involves a fourth-order system to further demonstrate the
effectiveness of the proposed data-driven event-triggering scheme. Consider a
linearized inverted pendulum system[35], whose state-space representation is
given by_

$\dot{x}(t)\\!=\\!\left[\begin{array}[]{cccc}\\!0&1&0&0\\!\\\
\\!0&0&{-m_{1}g}/{m_{2}}&0\\!\\\ \\!0&0&0&1\\!\\\ \\!0&0&{g}/{l}&0\\!\\\
\end{array}\right]\\!x(t)\\!+\\!\left[\begin{array}[]{cccc}\\!0\\!\\\
\\!{1}/{m_{2}}\\!\\\ \\!0\\!\\\ \\!{-1}/{(m_{2}l)}\\!\\\
\end{array}\right]\\!Kx(t)$

_where $m_{1}=1kg$ is the mass of pendulum bob, $m_{2}=10kg$ is the cart mass,
$l=3m$ is the length of the pendulum arm, and $g=10m/s^{2}$ is the
gravitational acceleration. The state variables
$[x_{1},~{}x_{2},~{}x_{3},~{}x_{4}]^{T}$ are the cart’s position, the cart’s
velocity, the pendulum angle, and the pendulum bob’s angle velocity. The
system matrices are assumed unknown, but we have $\rho=50$ measurements
$\\{\dot{x}(T_{i}),~{}x(T_{i}),~{}u(T_{i})\\}^{50}_{i=1}$ with the interval
satisfying $T_{i+1}-T_{i}=0.1$, where the input was sampled uniformly from
$u(t)\in[-1,~{}1]$. The disturbance $w(t)\in[-0.001,\,0.001]$ fulfills
Assumption 3 with $Q_{d}=-I$, $S_{d}=0$, and $R_{d}=0.001^{2}\rho I$
$(\rho=100)$._

We set the sampling interval $h=0.1$, triggering parameters $\sigma_{1}=0.01$,
$\sigma_{2}=0.01$, $\theta=2$, $\lambda=0.5$, delay bounds $\underline{d}=0$,
$\bar{d}=0.01$, and the matrix $B_{w}=0.01I$. Solving the LMIs (184) and (189)
in Theorem 5 with $\epsilon=2$, the controller gain and the triggering matrix
were co-designed as follows

$\displaystyle K$
$\displaystyle=[\begin{array}[]{cccc}2.4891&11.9277&297.9385&166.4980\\\
\end{array}]$ $\displaystyle\Omega$
$\displaystyle=\left[\begin{array}[]{cccc}19.8622&-49.6572&-4.7734&18.2200\\\
-49.6572&196.0430&11.0141&-61.6847\\\ -4.7734&11.0141&1.3070&-4.4237\\\
18.2200&-61.6847&-4.4237&20.7782\\\ \end{array}\right].$

We simulated system (3) under the triggering scheme in (47) with the initial
condition $x(0)=[0.98,~{}0,~{}0.2,~{}0]^{T}$, as well as the dynamic variable
$\eta(t)$, over the time interval $[0,\,30]$. The results in Fig. 5 show that
$x(t)$ and $\eta(\tau_{j}-d)$ approach zero as $t\to\infty$ under the designed
controller gain $K$ and the triggering matrix $\Omega$. In addition, under the
triggering scheme (47), _only_ $94$ out of 300 sampled data were transmitted
to the controller, corroborating the effectiveness of our data-driven event-
triggering scheme on the fourth-order system.

##  V Concluding Remarks

In this paper, we proposed a general dynamic event-triggering transmission
scheme for sample-data control systems with communication delays. As an
intermediate result of independent interest, we also provide a novel looped-
functional-based approach for event-triggered control with multiple advantages
(reduced computational complexity, reduced conservatism) over existing methods
in the literature. Both model- and data-based stability conditions were
developed using the looped-functional approach. Meanwhile, methods for co-
designing the controller gain and the triggering matrix were presented for
both cases of having a known or an unknown input matrix. Finally, two
numerical examples were provided to corroborate the role of additional prior
knowledge of the input matrix in reducing conservatism of our data-driven
stability conditions, as well as the merits and effectiveness of our methods.

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