# Greedy Synthesis of Event- and Self-Triggered Controls
with Control Lyapunov-Barrier Function

Masako Kishida *This work was supported by JST, PRESTO Grant Number
JPMJPR22C3, Japan.Masako Kishida is with the National Institute of
Informatics, Tokyo 101-8430, Japan<EMAIL_ADDRESS>
###### Abstract

This paper addresses the co-design problem of control inputs and execution
decisions for event- and self-triggered controls subject to constraints given
by the control Lyapunov function and control barrier function. The proposed
approach computes the control input in a way that allows for longer inter-
execution intervals, which distinguishes it from many existing event- and
self-triggered controllers or control Lyapunov-barrier function controllers.
The proposed approach guarantees lower bounds on the minimum inter-execution
times. The effectiveness of the proposed approach is demonstrated and compared
with existing approaches using a numerical example.

##  I INTRODUCTION

Control Barrier Function (CBF) [1] is a function used to design control inputs
to satisfy safety requirements. As the use of automatic control increases in
safety-critical systems, CBF is attracting much attention recently.
Applications of CBF can be found such as in robotics [2] and adaptive cruise
control [3]. Furthermore, CBF has been combined with signal temporal logic
[4], model predictive control [5] and extended to assuring risk-sensitive
safety [6], making it more versatile for a wider range of applications. On the
other hand, Control Lyapunov Function (CLF) [7, 8], an extension of the
Lyapunov function, has been widely used to design stabilizing controllers for
many different problems (see e.g., [9, 10, 11, 12]). After an integration of
CLF and CBF was proposed in [13, 14], it has been shown that CLF and CBF can
be combined to form a quadratic program (QP) for computing control inputs that
ensure safety while aiming at stability in [15, 16]. Since then, the CLF-CBF
QP approach has been used to solve various problems, such as autonomous
surface vehicles [17] and safe stabilization [18].

With the increasing prevalence of networked control systems, where the
communication bandwidth is shared with other tasks or batteries are used in
the system elements, it has become crucial to design systems that use
communication bandwidth and energy efficiently. Event- and self-triggered
control approaches are effective solutions to minimize unnecessary
communication and energy consumption for such systems [19, 20, 21].

Several approaches have been proposed for safety-critical systems using event-
or self-triggered strategies. An event-triggered control based on input-to-
state safe barrier functions by using a state feedback control law $u=k(x)$
was proposed in [22]. The approach is to bound the difference between the
current state and the state used to compute the control input, i.e., the state
at the previous execution time instance, to guarantee that the value of the
barrier function monotonically decreases. More recently, [23] proposed an
approach of event-triggered control for multi-agent systems with unknown
dynamics. It synthesizes an event-triggered control with an adaptive affine
dynamics that are updated based on the error states to estimate the real
system state. A combination of self-triggered control and CLF-CBF QP was
proposed in [24]; however, it does not seem to guarantee the existence of the
control input. Moreover, it possibly results in continuous control updates if
the optimal control input is achieved at the boundary of the QP.

The main contribution of this paper is to introduce approaches to co-designing
control inputs and execution time instances for event- and self- triggered
controls, with the aim of meeting the constraints given by the control
Lyapunov function and control barrier function while reducing the number of
executions. This is achieved by computing the control input so as to obtain
long inter-execution intervals in a greedy manner. The approach is different
from many existing event- and self-triggered controllers that rely on constant
feedback laws [20] or control Lyapunov-barrier function controllers that
account for only the cost of control inputs. It is also shown that the
proposed approach does not exhibit Zeno behavior and that the optimization
parameters appear in a lower bound on the minimum inter-execution time.

The rest of the paper is organized as follows. After introducing the notation,
system model, and basics of control Lyapunov-barrier function in Section II,
Section III presents the proposed event-triggered control approach. Section IV
discusses the extension to the self-triggered control approach. The
performances of those proposed controllers are illustrated and compared with
existing controllers in Section V, which is followed by conclusions in Section
VI.

##  II PRELIMINARIES

###  II-A Notation

The sets of real numbers, real vectors of length $n$, and real matrices of
size $n\times m$ are denoted by $\mathbb{R}$, $\mathbb{R}^{n}$, and
$\mathbb{R}^{n\times m}$, respectively. The sets of nonnegative numbers and
nonnegative integers are denoted by $\mathbb{R}_{\geq 0}$ and $\mathbb{N}$,
respectively.

$L_{f}V(x)$ denotes the Lie derivative of $V(x)$ along the vector field
$f(x)$, i.e. $L_{f}V(x)=\frac{\partial V(x)}{\partial x}f(x)$.

A continuous function $\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}$ is said to belong to class $\mathcal{K}$ if it is strictly increasing and
$\gamma(0)=0$. A continuous function $\alpha:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$ is said to belong to class
$\mathcal{K}_{\infty}$ if it belongs to class $\mathcal{K}$ and
$\alpha(r)\rightarrow\infty$ as $r\rightarrow\infty$.

###  II-B System Model

This paper deals with a nonlinear affine system in the form of

$\displaystyle\dot{x}=f(x)+g(x)u,\ x(0)=x_{0}\in\mathcal{C}$ (1)

where $x\in\mathcal{D}\subset\mathbb{R}^{n}$ and $u\in\mathbb{R}^{m}$ denote
the state and the control input of the system, respectively, and
$\mathcal{C}\subseteq\mathcal{D}$ is a safe set that is defined later. It is
assumed that $\mathcal{D}$ is bounded, and the functions $f(x)$ and $g(x)$ are
Lipschitz.

###  II-C Control Lyapunov Function

######  Definition II.1 (Control Lyapunov Function)

A positive definite function $V:\mathcal{D}\rightarrow\mathbb{R}_{\geq 0}$ is
called a Control Lyapunov Function (CLF) if it satisfies

$\displaystyle\inf_{u\in\mathcal{U}}L_{f}V(x)+L_{g}V(x)u\leq-\gamma(V(x))$ (2)

where $\gamma$ is a class $\mathcal{K}$ function.

The existence of a CLF guarantees asymptotic stabilization of the nonlinear
control system (1) with any Lipschitz continuous feedback controller $u(x)$
that satisfies (2) for all $x\in\mathcal{D}$ [25, 16].

###  II-D Control Barrier Function

Let define the safe set $\mathcal{C}$ as the superlevel set of a continuously
differentiable function
$h:\mathcal{D}\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$

$\displaystyle\mathcal{C}=\\{x\in\mathcal{D}:h(x)\geq 0\\}.$ (3)

######  Definition II.2 (Control Barrier Function)

A function $h:\mathcal{D}\rightarrow\mathbb{R}$ is called a Control Barrier
Function (CBF) if it satisfies

$\displaystyle\sup_{u\in\mathcal{U}}L_{f}h(x)+L_{g}h(x)u\geq-\alpha(h(x))$ (4)

for all $x\in\mathcal{D}$, where $\alpha$ is a class $\mathcal{K}_{\infty}$
function.

The existence of a CLF guarantees that the control system is safe [15].

###  II-E CLF-CBF-Based QP

Motivated by the results on CLF and CBF, it is of interest to obtain a
controller that satisfies

$\displaystyle L_{f}V(x)+L_{g}V(x)u\leq-\gamma(V(x)),$ (5) $\displaystyle
L_{f}h(x)+L_{g}h(x)u\geq-\alpha(h(x))$ (6)

so that it is a safe stabilizing controller.

To design such a controller, a CLF-CBF QP was constructed in [26, 15]:

$\displaystyle\begin{aligned}
\mathbf{v}^{*}(x)&=\operatorname*{arg\,min}_{\mathbf{v}=[u,\delta]^{\top}\in\mathbb{R}^{m+1}}\frac{1}{2}\mathbf{v}^{\top}Q\mathbf{v}+c^{\top}\mathbf{v}\\\
\text{s.t. }\ &\left[\begin{matrix}L_{g}V(x)&-1\\\
-L_{g}h(x)&0\end{matrix}\right]\mathbf{v}\leq\left[\begin{matrix}b_{\text{clf}}(x)\\\
b_{\text{cbf}}(x)\end{matrix}\right],\end{aligned}$ (7)

where

$\displaystyle\begin{aligned} b_{\text{clf}}(x)&=-L_{f}V(x)-\gamma(V(x)),\\\
b_{\text{cbf}}(x)&=L_{f}h(x)+\alpha(h(x)),\end{aligned}$ (8)

a positive definite matrix $Q\in\mathbb{R}^{m+1\times m+1}$ and
$c\in\mathbb{R}^{m+1}$ are weights and $\delta$ is a relaxation variable that
ensures the solvability of the QP. If $\delta$ is forced to be nonpositive,
then the existence of a feasible controller will guarantee the monotonic
decrease of the Lyapunov function.

##  III EVENT-TRIGGERED CONTROL

This section proposes a greedy event-triggered control with the control
Lyapunov-barrier function for the system (1).

###  III-A Event-triggered controller structure

The primary idea behind event-triggered control is to update the control input
only when necessary to achieve a specified performance condition, thereby
reducing the frequency of updates. In line with the standard structure of an
event-triggered controller, we consider the control inputs that are maintained
constant between successive event times, i.e.,

$\displaystyle u(t)$ $\displaystyle=u_{k},\ t\in[t_{k},t_{k+1}),$ (9)

where $u_{k}$ is the control input computed at time $t_{k}$, which is the time
instance when the control input is re-computed and the actuator signals are
updated. The time instance $t_{k}$ is determined by

$\displaystyle\begin{aligned} &t_{0}=0,\\\
&t_{k+1}=\inf\\{t\in\mathbb{R}:t>t_{k}\text{ and }\text{ trigger condition is
met}\\}.\end{aligned}$ (10)

###  III-B Greedy control update

Here, we introduce a greedy approach for computing the control input $u_{k}$
at trigger time $t_{k}$. To implement it into an event-triggered control, we
are interested in a control law that maximizes the inter-execution time. For
this purpose, we seek the control input that brings the state away from the
boundaries of the constraints (5), (6). This is achieved by maximizing the
slack variables $\rho_{1}$ and $\rho_{2}$ in the new constraints:

$\displaystyle\begin{aligned} L_{g}V(x)u+\rho_{1}&\leq b_{\text{clf}}(x),\\\
-L_{g}h(x)u+\rho_{2}&\leq b_{\text{cbf}}(x),\end{aligned}$ (11)

where $b_{\text{clf}}(x)$ and $b_{\text{cbf}}(x)$ are defined in (8).

Based on the constraints (11), we propose to modify the CLF-CBF QP in (7) as
follows:

$\displaystyle\begin{aligned}
\mathbf{v}^{*}(x)&=\operatorname*{arg\,min}_{\mathbf{v}=[u,\rho_{1},\rho_{2}]^{\top}\in\mathbb{R}^{m+2}}\frac{1}{2}\mathbf{v}^{\top}Q\mathbf{v}+c^{\top}\mathbf{v}\\\
\text{s.t. }\ &\left[\begin{matrix}L_{g}V(x)&1&0\\\ -L_{g}h(x)&0&1\\\
0&0&-1\end{matrix}\right]\mathbf{v}\leq\left[\begin{matrix}b_{\text{clf}}(x)\\\
b_{\text{cbf}}(x)\\\
-\varepsilon_{\text{cbf}}\end{matrix}\right],\end{aligned}$ (12)

where a positive semidefinite matrix $Q\in\mathbb{R}^{m+2\times m+2}$ and
$c\in\mathbb{R}^{m+2}$ are weights, $\rho_{1}$ and $\rho_{2}$ are slack
variables, and $\varepsilon_{\text{cbf}}>0$ is a constant (design parameter)
that forces the states to be away from the boundary. This is still QP and the
solvability of the QP is still ensured as long as $\rho_{1}$ is not
constrained. Let
$[u^{*}(x),\rho_{1}^{*}(x),\rho_{2}^{*}(x)]^{\top}=\mathbf{v}^{*}(x)$.

Typically, a small norm control input $u$ is desired to minimize the control
effort while large $\rho_{1}$ and $\rho_{2}$ are desired to maximize the
inter-execution time. Hence, a possible choice for $Q$ and $q$ is

$\displaystyle Q=\left[\begin{matrix}w_{1}&0&0\\\ 0&0&0\\\ \ 0&0&0\
\end{matrix}\right],\
c=\left[\begin{matrix}0&-w_{2}&-w_{3}\end{matrix}\right]$ (13)

where $w_{1}\in\mathbb{R}^{m\times m}$ is positive definite, and
$w_{2},w_{3}\geq 0$.

With (12), the proposed controller implements the control input $u_{k}$
defined by

$\displaystyle
u_{k}=u^{*}(x_{k})=\left[\begin{matrix}1&0&0\end{matrix}\right]^{\top}\mathbf{v}^{*}(x_{k}),$
(14)

where $x_{k}=x(t_{k})$.

###  III-C Trigger conditions

For the states to remain in the safe region, the safety constraint (6) should
be always satisfied. However, the satisfaction of the stability constraint (5)
cannot be guaranteed together with the satisfaction of (6) in general. This is
the same for the proposed controller. Yet, we do our best to minimize the time
in which (5) is violated.

Define

$\displaystyle p(x)$ $\displaystyle=p_{k}(x),\ t\in[t_{k},t_{k+1}),$ (15)
$\displaystyle q(x)$ $\displaystyle=q_{k}(x),\ t\in[t_{k},t_{k+1})$ (16)

where

$\displaystyle p_{k}(x)$ $\displaystyle=-L_{f}V(x)-L_{g}V(x)u_{k},$ (17)
$\displaystyle q_{k}(x)$ $\displaystyle=L_{g}h(x)u_{k}+b_{\text{cbf}}(x).$
(18)

Note that the time derivative of the Lyapunov function is
$\dot{V}(x)=-p_{k}(x)$, thus $p_{k}(x)\geq 0$ is desired for the stability,
while $q_{k}(x)\geq 0$ is desired for the safety.

We set the trigger condition in (10) as

  1. 1.

if $p_{k}(x_{k})\geq\varepsilon_{\text{clf}}$:

$\displaystyle p_{k}(x)=0\text{ or }q_{k}(x)=0$ (19)

  2. 2.

else:

$\displaystyle q_{k}(x)=0\text{ or }t=t_{k}+\tau_{bd}$ (20)

where $\varepsilon_{\text{clf}}>0$ and $\tau_{bd}>0$ are design parameters.

The first case is that, if the time-derivative of the Lyapunov function is
sufficiently negative at the time of update $t_{k}$, then the next update time
is when either the safety constraint (5) is satisfied with equality or the
time-derivative of Lyapunov function becomes zero. This guarantees the
satisfaction of both safety and stability between $t_{k}$ and $t_{k+1}$.

The second case is that, if the time-derivative of the Lyapunov function is
close to zero or positive at the time of update $t_{k}$, then we compromise
the controller design only focusing on the safety constraint. The next update
time is when the safety constraint (5) is satisfied with equality or small
time $\tau_{bd}$ passes, whichever occurs first. This limits the duration of
time during which the stability constraint is violated to $\tau_{bd}$ with the
same control input.

###  III-D Lower bound on minimum inter-execution time

This subsection shows that the events cannot be triggered an infinite number
of times in any finite time period with the proposed controller, i.e., the
proposed controller is Zeno-free under mild conditions.

Define the inter-execution time

$\displaystyle\tau_{k}=t_{k+1}-t_{k},\ k\in\mathbb{N}.$ (21)

It is of interest to show the existence of a lower bound $\tau^{*}$ such that
$\tau_{k}\geq\tau^{*}$ for all $k\in\mathbb{N}$.

###### Assumption III.1

We assume the followings hold on $\mathcal{D}$:

  * •

$\|f(x)+g(x)u\|$ is bounded above, and

  * •

There exists a Lipschitz constant $L_{\text{clf}}>0$ for $p(x)$

  * •

There exists a Lipschitz constant $L_{\text{cbf}}>0$ for $q(x)$

If the problem is considered in a finite time horizon, the first assumption is
sufficient.

###### Theorem III.2

Under Assumption III.1, for the time sequence (10) for the system (1) with the
event-triggered controller (9), (14) with the trigger condition (19)-(20),
there exists $\tau^{*}>0$ such that $\tau_{k}\geq\tau^{*}$ for all
$k\in\mathbb{N}$.

###### Proof:

First, we consider if $p_{k}(x_{k})\geq\varepsilon_{\text{clf}}$, then how
long it takes to achieve

$\displaystyle p_{k}(x)=0$ (22)

for the first time after $t_{k}$. Let this time instance be
$\bar{t}_{\text{clf}}$ and the corresponding state be $\bar{x}_{\text{clf}}$.

For this, we first show that

$\displaystyle\|\bar{x}_{\text{clf}}-x_{k}\|\geq\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}}.$
(23)

Clearly,

$\displaystyle
p_{k}(x_{k})-p_{k}(\bar{x}_{\text{clf}})\geq\varepsilon_{\text{clf}},$ (24)

and

$\displaystyle L_{\text{clf}}\|x_{k}-\bar{x}_{\text{clf}}\|\geq
p_{k}(x_{k})-p_{k}(\bar{x}_{\text{clf}}).$ (25)

Together, it follows that

$\displaystyle\|\bar{x}_{\text{clf}}-x_{k}\|\geq\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}}.$
(26)

Next, we show that the time difference $\bar{t}_{\text{clf}}-t_{k}$ is lower
bound away from zero. By the fundamental theorem of calculus, we have

$\displaystyle\begin{aligned}
\|\bar{x}_{\text{clf}}-x_{k}\|&=\left\|\int_{t_{k}}^{\bar{t}_{\text{clf}}}\dot{x}(\tau)d\tau\right\|\\\
&\leq\sup\|f(x)+g(x)u\|(\bar{t}_{\text{clf}}-t_{k}).\end{aligned}$ (27)

Because $\|f(x)+g(x)u\|$ is bounded above, there exists $M>0$ such that
$M=\sup\|f(x)+g(x)u\|$, then it follows that

$\displaystyle\bar{t}_{\text{clf}}-t_{k}\geq\frac{1}{M}\|\bar{x}_{\text{clf}}-x_{k}\|\geq\frac{\varepsilon_{\text{clf}}}{ML_{\text{clf}}}.$
(28)

Thus, at least the time interval of
$\frac{\varepsilon_{\text{clf}}}{ML_{\text{clf}}}$ takes to achieve (22).

Similarly, we consider how long it takes to achieve

$\displaystyle q_{k}(x)=0$ (29)

for the first time after $t_{k}$. Let this time instance
$\bar{t}_{\text{cbf}}$ and the corresponding state $\bar{x}_{\text{cbf}}$.

Using the fact that
$q_{k}(x_{k})\geq\rho_{2}^{*}(x_{k})\geq\varepsilon_{\text{cbf}}$, we can show
that

$\displaystyle\|\bar{x}_{\text{cbf}}-x_{k}\|\geq\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}}$
(30)

and then

$\displaystyle\bar{t}_{\text{cbf}}-t_{k}\geq\frac{1}{M}\|\bar{x}_{\text{cbf}}-x_{k}\|\geq\frac{\varepsilon_{\text{cbf}}}{ML_{\text{cbf}}}.$
(31)

In summary, the inter-execution time is lower bounded by
$\tau^{*}=\min(\frac{\varepsilon_{\text{clf}}}{ML_{\text{clf}}},\frac{\varepsilon_{\text{cbf}}}{ML_{\text{cbf}}},\
\tau_{bd})$, which is strictly positive. This completes the proof. ∎

Here, we see the parameter $\varepsilon_{\text{cbf}}>0$ in (12) can be used as
a parameter to tune the inter-execution time.

###  III-E Special cases

####  III-E1 Feasibility is guaranteed

Suppose that the feasibility of the constraints (5) and (6) is guaranteed for
all $x\in\mathcal{D}$ with some margins, i.e., there exist $s_{1},s_{2}>0$
such that for all $x\in\mathcal{D}$, there exists an control input $u$ that
depends on $x$ that satisfies

$\displaystyle\begin{aligned} L_{g}V(x)u+s_{1}&\leq b_{\text{clf}}(x),\\\
-L_{g}h(x)u+s_{2}&\leq b_{\text{cbf}}(x).\end{aligned}$ (32)

Then, we may consider not only forcing the relaxation variable $\delta$ to be
0 in (7), but securing certain distances from the boundaries of the
constraints. This allows us to add a constraint $\rho_{1}\geq 0$ in (12) and
the resulting QP is:

$\displaystyle\begin{aligned}
\mathbf{v}^{*}(x)&=\operatorname*{arg\,min}_{\mathbf{v}=[u,\rho_{1},\rho_{2}]^{\top}\in\mathbb{R}^{m+2}}\frac{1}{2}\mathbf{v}^{\top}Q\mathbf{v}+c^{\top}\mathbf{v}\\\
\text{s.t. }\ &\left[\begin{matrix}L_{g}V(x)&1&0\\\ -L_{g}h(x)&0&1\\\
0&-1&0\\\
0&0&-1\end{matrix}\right]\mathbf{v}\leq\left[\begin{matrix}b_{\text{clf}}(x)\\\
b_{\text{cbf}}(x)\\\ -\varepsilon_{1}\\\
-\varepsilon_{2}\end{matrix}\right],\end{aligned}$ (33)

where $\varepsilon_{1}\in(0,s_{1}),\varepsilon_{2}\in(0,s_{2})$ are constants,
a positive semidefinite matrix $Q\in\mathbb{R}^{m+2\times m+2}$, and
$c\in\mathbb{R}^{m+2}$ are the weights. In this case, the trigger condition
(19)-(20) can be combined and modified as:

$\displaystyle(L_{g}V(x)u-b_{\text{clf}}(x))(L_{g}h(x)u+b_{\text{cbf}}(x))=0$
(34)

to guarantee both stability and safety.

####  III-E2 Control input is not penalized

In addition that the feasibility is guaranteed, if the control input $u$ is
not penalized as in other event-triggered controls, then, the problem
simplifies to solving a linear program:

$\displaystyle\begin{aligned}
u^{*}(x)&=\operatorname*{arg\,min}_{u\in\mathbb{R}^{m}}\left(-w_{2}L_{g}V(x)-w_{3}L_{g}h(x)\right)u\\\
\text{s.t. }\ &\left[\begin{matrix}L_{g}V(x)\\\
-L_{g}h(x)\end{matrix}\right]u\leq\left[\begin{matrix}b_{\text{clf}}(x)-\varepsilon_{1}\\\
b_{\text{cbf}}(x)-\varepsilon_{2}\end{matrix}\right],\end{aligned}$ (35)

where, again, $\varepsilon_{1}\in(0,s_{1}),\varepsilon_{2}\in(0,s_{2})$ are
constants, using some weights $w_{2},w_{3}\geq 0$.

##  IV Self-triggered control

This section develops a greedy self-triggered control based on the results in
Section III.

###  IV-A Self-triggered controller structure

As in Section III, let $t_{k}$ be the triggering time instance. Self-triggered
control computes the control input $u_{k}$ as well as the next time instance
$t_{k+1}$ at execution time $t_{k}$. Similar to the event-triggered control
(9), the control input $u_{k}$ remains constant in between $t_{k}$ and
$t_{k+1}$. Unlike the event-triggered condition, however, no sampling or
computation is required between $t_{k}$ and $t_{k+1}$.

As for the event-triggered control, the controller implements the control
input $u_{k}$ in (14) by solving (12).

The next execution time $t_{k+1}$ is computed based on the measured state
$x(t_{k})=x_{k}$ at $t_{k}$:

$\displaystyle\begin{aligned} &t_{0}=0,\\\
&t_{k+1}=t_{k}+\Gamma(x_{k}),\end{aligned}$ (36)

where the map $\Gamma:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ determines
the triggering time $t_{k+1}$ as a function of the state $x_{k}$ at the time
$t_{k}$. Thus, the inter-execution time is given by $\Gamma(x)$, i.e.,
$\tau_{k}=\Gamma(x_{k})$.

In the following, we present approaches to how to design the map $\Gamma(x)$.

###  IV-B Computing the next execution time instance

In Section III, the trigger condition (19)-(20) is designed to satisfy the
safety constraint (6) and minimize the violation of the stability constraint
(5). Here, again, we design the map $\Gamma$ that satisfies the safety
constraint (6) all the time, while allowing the violation of the stability
constraint:

$\displaystyle\begin{aligned} &\Gamma(x_{k})\leq\sup\\{t>t_{k}:\\\ &\ \text{
if }p_{k}(x_{k})\geq\varepsilon_{\text{clf}}:\\\ &\qquad p_{k}(x(\tau))\geq
0\text{ and }q_{k}(x(\tau))\geq 0\text{ for all }\tau\leq t\\\ &\ \text{ else:
}\\\ &\qquad p_{k}(x(\tau))\geq 0\text{ for all }\tau\leq t\text{ and }t\leq
t_{k}+\tau_{bd}\\}\\\ &\ -t_{k}.\end{aligned}$ (37)

Ideally, we would like to find $t$ that achieves an equality in (37). However,
it is difficult in general for nonlinear systems, thus, we aim at finding a
lower bound by revising the approach in Section III-D.

Again, we assume that Assumption III.1 is satisfied. Then, we may use the
following map:

###### Theorem IV.1

Under Assumption III.1,

$\displaystyle\Gamma(x_{k})=\begin{cases}\min\left(\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}M_{k}},\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}M_{k}}\right),\text{
if }p_{k}(x_{k})\geq\varepsilon_{\text{clf}}\\\
\min\left(\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}M_{k}},\tau_{bd}\right),\text{
otherwise },\\\ \end{cases}$ (38)

where

$\displaystyle
M_{k}=\sup_{t\in\left[t_{k},t_{k}+\Gamma(x_{k})\right)}\|f(x)+g(x)u_{k}\|$
(39)

satisfies (37).

###### Proof:

Here, again note that after solving (12), it is guaranteed that

$\displaystyle
q_{k}(x_{k})\geq\rho_{2}^{*}(x_{k})\geq\varepsilon_{\text{cbf}}.$ (40)

For the first case of $p_{k}(x_{k})\geq\varepsilon_{\text{clf}}$, we show

$\displaystyle\Gamma(x_{k})=\min\left(\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}M_{k}},\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}M_{k}}\right),$
(41)

satisfies (37). By the fundamental theorem of calculus, the equation (41)
implies that for $t\in[t_{k},t_{k}+\Gamma(x_{k}))$,

$\displaystyle\begin{aligned}
\|x-x_{k}\|&\leq\int_{t_{k}}^{t}\|f(x)+g(x)u_{k}\|dt\\\ &\leq(t-t_{k})M_{k}\\\
&\leq(t-t_{k})\frac{\min\left(\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}},\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}}\right)}{\Gamma(x_{k})}\\\
&\leq\min\left(\frac{\varepsilon_{\text{clf}}}{L_{\text{clf}}},\frac{\varepsilon_{\text{cbf}}}{L_{\text{cbf}}}\right)\\\
\Leftrightarrow\
&\begin{cases}\varepsilon_{\text{clf}}-L_{\text{clf}}\|x-x_{k}\|\geq 0,\\\
\varepsilon_{\text{cbf}}-L_{\text{cbf}}\|x-x_{k}\|\geq
0.\end{cases}\end{aligned}$ (42)

On the other hand,

$\displaystyle\begin{aligned} \varepsilon_{\text{clf}}-p_{k}(x)\leq
p_{k}(x_{k})-p_{k}(x)\leq L_{\text{clf}}\|x-x_{k}\|,\\\
\varepsilon_{\text{cbf}}-q_{k}(x)\leq q_{k}(x_{k})-q_{k}(x)\leq
L_{\text{cbf}}\|x-x_{k}\|.\end{aligned}$ (43)

Hence, it follows that

$\displaystyle p_{k}(x)\geq 0,\ q_{k}(x)\geq 0.$ (44)

The second case is clear from the above discussions. Together, it completes
the proof. ∎

The difference between this map $\Sigma$ and the lower bound $\tau^{*}$ in the
event-triggered control is only the upper bound on the norm of $f(x)+g(x)u$.
Although a uniform $M$ may be used to design a constant $\Gamma=\tau^{*}$,
such a law will shorten the inter-execution time because $M$ is likely to be
much larger than $M_{k}$. However, one difficulty of implementing this
approach is actually in the computation of $M_{k}$ in (47) where $M_{k}$
appears in both sides of the equation. To compute exact $M_{k}$, it is
required to compute the evolution of (1) starting at $t_{k}$ by gradually
increasing the time duration. In the actual implementation, this might not be
a big problem because implementation is done digitally, which we discuss next.

###  IV-C Digital implementation

Similarly to [20], we now consider the following discrete-time versions of
$p_{k}(x)$ and $q_{k}(x)$ based on a sampling time $\Delta>0$, which are
defined by

$\displaystyle\begin{aligned} \bar{p}_{k}(n)&=p_{k}(x(t_{k}+n\Delta)),\\\
\bar{q}_{k}(n)&=q_{k}(x(t_{k}+n\Delta)).\end{aligned}$ (45)

Let $\tau_{\min}$ and $\tau_{\max}$ be design parameters and let
$N_{\min}=\lfloor\tau_{\min}/\Delta\rfloor$,
$N_{\max}=\lfloor\tau_{\max}/\Delta\rfloor$.

Then, by choosing $\tau_{bd}=\Delta$, the map can be simplified as

$\displaystyle\Gamma(x_{k})=\max\\{\tau_{\min},n(x_{k})\Delta\\},$ (46)

where

$\displaystyle\begin{aligned} n(x)=\max_{n\in\mathbb{N}}\\{n\leq
N_{\max}:\bar{p}_{k}(m)&\geq 0\text{ and }\bar{q}_{k}(m)\geq 0,\\\ &\qquad
m=1,\cdots,n\\}.\end{aligned}$ (47)

This satisfies

$\displaystyle\bar{p}_{k}(n)\geq 0\text{ and }\bar{q}_{k}(n)\geq 0,\ \forall
n\in\left[0,\left\lceil\frac{t_{k+1}-t_{k}}{\Delta}\right\rceil\right)$ (48)

and $n(x)\in[N_{\min},N_{\max}]$.

With a smaller choice of $\Delta$, the stability constraint is more strictly
forced, i.e., while the stability constraint is violated, the control input is
updated every sampling time. Such a choice of $\Delta$ may result in an
increase in the number of execution. Of course, it is possible to use
different values for $\tau_{bd}$ and $\Delta$ with a slight modification.

We may simply choose $\tau_{\min}=0$ and a sufficiently large $\tau_{\max}$.
However, the value of $\tau_{\max}$ enforces the robustness of the
implementation and limits the computational complexity [20].

##  V NUMERICAL EXAMPLE

This section demonstrates the effectiveness of the proposed approach using an
example of a double integrator.

Let $x=\left[\begin{matrix}x_{1}&x_{2}\end{matrix}\right]^{\top}$, where
$x_{1}$ is the position and $x_{2}$ is the velocity. The dynamics of a double
integrator is given as follows:

$\displaystyle\dot{x}$ $\displaystyle=\left[\begin{matrix}0&1\\\
0&0\end{matrix}\right]x+\left[\begin{matrix}0\\\ 1\end{matrix}\right]u.$ (49)

The control Lyapunov and control barrier functions are selected as

$\displaystyle\begin{aligned} V(x)&=x_{1}^{2}+x_{1}x_{2}+x_{2}^{2},\\\
h(x)&=(x_{1}-0.5)^{2}+(x_{2}+0.5)^{2}-0.3^{2}.\end{aligned}$ (50)

Moreover, the $\alpha$ an $\gamma$ functions are chosen to be identity maps
and the parameters $\varepsilon_{\text{clf}}=\varepsilon_{\text{cbf}}=0.1$,
$\tau_{bd}=0.5$, $\Delta=0.2$, $\tau_{\min}=0$ and $\tau_{\max}=4$ are
selected. The weights for QP in (12) are selected as

$\displaystyle Q=\left[\begin{matrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}\right],\
c=\left[\begin{matrix}0&-1&0\end{matrix}\right]^{\top}.$ (51)

Here, the performances of the following five controllers are compared for the
duration of time 15, starting at $x_{0}=[1,1]$.

  * •

Greedy ET: the controller that solves (12) and implements (14) when the
trigger condition (19) or (20) is met

  * •

Greedy ST: the controller that solves (12) and implements (14) at time
instances determined by (36) and (46)

  * •

Greedy: the controller that solves (12) and implements (14) continuously

  * •

CLF-CBF QP: CLF-CBF QP controller in [16] with $H(x)=2$, $p=1$

  * •

SF: a standard state-feedback controller with the controller gain
$K=\left[-0.5,-1\right]$, which corresponds to the case of weight for the
states is $\left[\begin{matrix}0.25&-0.5\\\ -0.5&0\end{matrix}\right]$ and
weight for the control input is $1$, thus $P=\left[\begin{matrix}1&0.5\\\
0.5&1\end{matrix}\right]$ in the algebraic Riccati equation. No constraints
are considered.

Figure 1: Phase portrait: the grey region indicates the unsafe region,
$h(x)<0$ Figure 2: Control inputs Figure 3: Lyapunov function values,
$V(x)=x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}$

(a) Stability constraint used for trigger condition, $L_{f}V(x)+L_{g}V(x)u$.
The values are desired to be negative.

(b) Safety constraint used for trigger condition, $L_{g}h(x)u+b_{cbf}(x)$. The
values are desired to be nonnegative.

Figure 4: Trigger conditions

Figure 1 shows the phase portraits for the five controllers. It is observed
that the trajectory with SF goes into the unsafe region, which motivates us to
use the barrier function to remain in the safe region. Also, the trajectories
of Greedy and CLF-CBF-QP are close to each other, but the state of CLF-CBF-QP
is further from the origin at the end of the simulation compared with the
other four methods. Moreover, both trajectories of Greedy ET and Greedy SF
take longer paths compared with other methods.

Figure 2 shows the control input trajectories. It can be seen that Greedy ET
and Greedy ST do not require frequent control updates. In fact, the numbers of
control updates for Greedy ET and Greedy ST were 24 and 26, respectively. With
Greedy ST, a smaller sampling time $\Delta$ forces the derivative of the
Lyapunov function to be negative more strictly thus tends to increase the
update frequency.

Figure 3 shows the Lyapunov function values. Both Greedy ET and Greedy ST
admit increases of the Lyapunov function values for certain periods. However,
those trajectories approach zero quickly, while CLF-CBF-QP is still away from
zero at the end of the simulation. Note that the original CLF-CBF-QP
controller also allows increases of the Lyapunov function values to guarantee
the feasibility of the optimization problem [16].

Figure 4 shows the trajectories of values used for triggers. Because Greedy ET
and Greedy ST compromised the stability constraints for the sake of reducing
the update frequencies, Figure 4 indicates the values of
$L_{f}V(x)+L_{g}V(x)u$ go above zero sometimes. On the other hand, the safety
constraints are always satisfied by all the controllers except for the SF that
ignored the existence of unsafe region Figure 4. In Figure 4, we also observe
that the trajectory $L_{g}h(x)u+b_{cbf}(x)$ of CLF-CBF-QP stays near zero
around time 2. Thus, we cannot implement an event-triggered strategy with CLF-
CBF-QP because the trigger condition is violated or close to violate already
at the time of the update.

##  VI CONCLUSIONS

In this paper, we have presented a set of greedy approaches for synthesizing
event-triggered and self-triggered controls with the control Lyapunov-Barrier
function. Our proposed approach computes each control input to maximize the
distance from the safety boundary, which is a departure from existing
approaches to the control Lyapunov-Barrier function. By doing so, our approach
ensures a positive lower bound on the minimum inter-execution time, while also
maintaining the safety of the control system and reducing the frequency of
control input updates and/or samplings. This is particularly beneficial in the
context of networked control systems, where safety is of paramount concern.
The effectiveness of our proposed approach has been illustrated through a
numerical example, which highlights its potential for real-world
implementation.

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