1

Traffic Models of Periodic Event-Triggered Control
Systems

arXiv:1711.03599v1 [] 9 Nov 2017

Anqi Fu, and Manuel Mazo, Jr.

Abstract—Periodic event-triggered control (PETC) [13] is a
version of event-triggered control (ETC) that only requires to
measure the plant output periodically instead of continuously. In
this work, we present a construction of timing models for these
PETC implementations to capture the dynamics of the traffic they
generate. In the construction, we employ a two-step approach.
We first partition the state space into a finite number of regions.
Then in each region, the event-triggering behavior is analyzed
with the help of LMIs. The state transitions among different
regions result from computing the reachable state set starting
from each region within the computed event time intervals.
Index Terms—systems abstractions; periodic event-triggered
control; LMI; formal methods; reachability analysis.

I. I NTRODUCTION
Wireless networked control systems (WNCS) are control
systems that employ wireless networks as feedback channels.
In such systems, the physically distributed components are
co-located with their own wireless nodes and communicate
via a wireless network. These components can be designed
with great mobility once the nodes are supported by batteries.
Besides, each component can be established and updated
easily. Therefore, WNCS have great adaptability on obtaining
different control objectives and have been attracting much
attention. However, there are two major issues that must be
considered while designing such a system: limited bandwidth
and energy supply.
Most often, control tasks are designed to be executed
periodically. This periodic strategy, also named time-triggered
control (TTC), does not regard the system’s current state and
thus may waste bandwidth and energy. Alternatively, eventtriggered control (ETC) strategies are proposed to reduce
bandwidth occupation, see e.g. [5], [17], [19], [22], [25], [26],
and references therein. In ETC, the control tasks only execute
when necessary, e.g. when some pre-designed performance indicator is about to be violated. Thus the system is tightfisted in
communication. However, to validate the pre-designed eventtriggering conditions, sensors are required to sample the plant
output continuously. This continuous monitoring can consume
large amounts of energy. To reduce this energy consumption,
naturally one may want to replace the continuously sampling
by a discrete time sampling.
When applying discrete time sampling, to compensate the
delay caused by the discretization, one can either design a
The authors are with the Delft Center for Systems and Control, Delft University of Technology, Delft 2628 CD, The Netherlands e-mail: A.Fu-1,
M.Mazo@tudelft.nl.
This work is partly funded by China Scholarship Council (CSC).

stricter event-triggering condition based on the system dynamics, as e.g. [18]; or modify the Lyapunov function, as e.g.
[13]. In [13], Heemels et. al. present a periodic event-triggered
control (PETC) mechanism. In a PETC implementation, the
sensors are only required to measure the plant output and
validate the event conditions periodically. Only when some
pre-designed conditions are satisfied, fresh measurements are
employed to recompute the controller output. Therefore, PETC
enjoys the benefits of both cautious communication and discrete time measurement. Compared to [18], the event conditions can be less conservative to further reduce communications. Thus the energy consumed and bandwidth occupied are
reduced. Furthermore, the transmissions of the control input
from the controller to the plant are also included in the PETC
mechanism.
To further reduce the resource consumption and to fully
extract the potential gains from ETC, one can also consider
scheduling approaches. By efficiently scheduling listening
times on wireless communications and medium access time in
general, the energy consumption in a WNCS can be reduced
and bandwidth can be more efficiently reused. To enable
such scheduling, a model for the traffic generated by ETC
is required. In [16], Kolarijani and Mazo propose a type of
approximate power quotient system, to derive models that
capture the timing behaviors of ETC systems applying the
triggering mechanism from [22]. They first partition the state
space into finite cones. In each cone, they analyze the timing
behavior by over-approximation methods (see e.g. [3], [4], [6],
[11], [14], [20], [21]), linear matrix inequality (LMI) methods,
and reachability analysis (see e.g. [1] and [2]).
Similarly, in order to fully extract the potential gains from
PETC with scheduling approaches, a model for the traffic
generated by PETC is necessary. In this work, we present a
construction of the timing models of the PETC implementations from [13]. First of all, we modify the PETC mechanism
by giving an upper bound time such that if no event happens
within that interval, the system will be forced to generate an
event by the end of it. When constructing the models, the
approach has two steps. We first divide the state space into
a finite number of partitions. For a 2-dimensional system, the
partition looks like a dartboard. Then we construct a set of
LMIs to compute the output map. Transition relations among
different regions are derived by computing the reachable state
set starting from each region. Compared with the work from
[9], we do not require that the perturbation should vanish as
the state converges. Instead, we only assume the perturbation
to be both L2 and L∞ .
This paper is organized as follows. Section II presents the

2

necessary notation and definitions. The problem to be solved
is defined in Section III. Section IV shows all the details to
construct a power quotient system to model the traffic of a
centralized PETC implementation. A numerical example is
shown in Section V. Section VI summarizes the contributions
of this paper and discusses future work. To ease the readability,
the proofs are collected in the Appendix.
II. N OTATION AND PRELIMINARIES
We denote the n-dimensional Euclidean space by Rn , the
+
positive real numbers by R+ , by R+
0 = R ∪ {0}. The natural
numbers including zero is denoted by N. When zero is not
included, we denote the natural numbers as N+ . IN+ is the
set of all closed intervals [a, b] such that a, b ∈ N+ and a ≤ b.
For any set S, 2S denotes the set of all subsets of S, i.e. the
power set of S. Mm×n and Mn are the set of all m × n real
valued matrices and the set of all n × n real-valued symmetric
matrices respectively. A symmetric matrix M ∈ Rn×n is said
to be positive (negative) definite, denoted by M  0 (M ≺ 0),
whenever xT M x > 0 (xT M x < 0) for all x 6= 0, x ∈ Rn .
M  0 (M  0) means M is a positive (negative) semidefinite matrix. When Q ⊆ Z × Z is an equivalence relation
on a set Z, [z] denotes the equivalence class of z ∈ Z and
Z/Q denotes the set of all equivalence classes. For a locally
integrable
signal w: R+ → Rn , we denote by kwkL2 =
qR
∞
|w(t)|2 dt its L2 -norm, kwkL∞ = supt≥0 kw(t)k < ∞
0
its L∞ -norm. Furthermore, we define the space of all locally
integrable signals with a finite L2 -norm as L2 , the space of
all signals with a finite L∞ -norm as L∞ .
Now we review some notions from the field of system
theory.
Definition 2.1: (Metric)[7] Consider a set T , d : T × T →
R ∪ {+∞} is a metric (or a distance function) if the following
three conditions are satisfied ∀x, y, z ∈ T :
• d(x, y) = d(y, x);
• d(x, y) = 0 ↔ x = y;
• d(x, y) ≤ d(x, z) + d(y, z).
The ordered pair (T, d) is said to be a metric space.
Definition 2.2: (Hausdorff distance)[7] Assume X and Y
are two non-empty subsets of a metric space (T, d). The
Hausdoorff distance dH (X, Y ) is given by:


max sup inf d(x, y), sup inf d(x, y) .
(1)
x∈X y∈Y

y∈Y x∈X

Definition 2.3: (System)[23] A system is a sextuple
(X, X0 , U, −→, Y, H) consisting of:
• a set of states X;
• a set of initial states X0 ⊆ X;
• a set of inputs U ;
• a transition relation −→⊆ X × U × X;
• a set of outputs Y ;
• an output map H : X → Y .
The term finite-state (infinite-state) system indicates X is a
finite (an infinite) set. For a system, if the cardinality of U is
smaller than or equal to one, then this system is said to be
autonomous.

Definition 2.4: (Metric system)[23] A system S is said to
be a metric system if the set of outputs Y is equipped with a
metric d : Y × Y → R+
0.
Definition 2.5: (Approximate simulation relation)[23] Consider two metric systems Sa and Sb with Ya = Yb , and let
 ∈ R+
0 . A relation R ⊆ Xa × Xb is an -approximate
simulation relation from Sa to Sb if the following three
conditions are satisfied:
• ∀xa0 ∈ Xa0 , ∃xb0 ∈ Xb0 such that (xa0 , xb0 ) ∈ R;
• ∀(xa , xb ) ∈ R we have d (Ha (xa ), Hb (xb )) ≤ ;
0
• ∀(xa , xb ) ∈ R such that (xa , ua , xa ) ∈−→ in Sa implies
a
0
∃(xb , ub , xb ) ∈−→ in Sb satisfying (x0a , x0b ) ∈ R.
b

We denote the existence of an -approximate simulation
relation from Sa to Sb by Sa S Sb , and say that
Sb -approximately simulates Sa or Sa is -approximately
simulated by Sb . Whenever  = 0, the inequality
d(Ha (xa ), Hb (xb )) ≤  implies Ha (xa ) = Hb (xb ) and the
resulting relation is called an (exact) simulation relation.
We introduce a notion of power quotient system and corresponding lemma for later analysis.
Definition 2.6: (Power quotient system)[16] Let S =
(X, X0 , U, −→, Y, H) be a system and R be an equivalence
relation on X. 
The power quotient of S by R, denoted
 by S/R ,
is the system

X/R , X/R,0 , U/R , −→, Y/R , H/R
/R

consisting

of:
•
•
•
•

X/R = X/R;

X/R,0 = x/R ∈ X/R |x/R ∩ X0 6= ∅ ;
U
 /R = U ; 
x/R , u, x0/R

∈−→ if ∃(x, u, x0 ) ∈−→ in S with x ∈
/R

x/R and x0 ∈ x0/R ;
Y
• Y/R ⊂ 2 ;
 S
• H/R x/R =
x∈x/R H(x).
Lemma 2.7: [Lemma 1 in [16]] Let S be a metric system,
R be an equivalence relation on X, and let the metric system
S/R be the power quotient system of S by R. For any

(2)
≥
max
d H(x), H/R x/R ,
x ∈ x/R
x/R ∈ X/R
with d the Hausdorff distance over the set 2Y , S/R approximately simulates S, i.e. S S S/R .
The definition of Minkowski addition is introduced here for
the computation of the reachable sets.
Definition 2.8: (Minkowski addition) The Minkowski addition of two sets of vectors A and B in Euclidean space is
formed by adding each vector in A to each vector in B:
A ⊕ B = {a + b|a ∈ A, b ∈ B},
where ⊕ denotes the Minkowski addition.
III. P ROBLEM DEFINITION
The centralized PETC presented in [13] is reviewed here.
Consider a continuous linear time-invariant (LTI) plant of the
form:
(
ξ˙p (t) = Ap ξp (t) + Bp v̂(t) + Ew(t)
(3)
y(t) = Cp ξp (t),

3

where ξp (t) ∈ Rnp denotes the state vector
y(t) ∈ Rny denotes the plant output vector,
denotes the input applied to the plant, w(t) ∈
the perturbation. The plant is controlled by a
controller, given by:
(
ξc (tk+1 ) = Ac ξc (tk ) + Bc ŷ(tk )

of the plant,
v̂(t) ∈ Rnv
Rnw denotes
discrete-time

v(tk ) = Cc ξc (tk ) + Dc ŷ(tk ),

(4)

where ξc (tk ) ∈ Rnc denotes the state vector of the controller,
v(tk ) ∈ Rnv denotes the controller output vector, and ŷ(tk ) ∈
Rny denotes the input applied to the controller. A periodic
sampling sequence is given by:
Ts := {tk |tk := kh, k ∈ N},

(5)

where h > 0 is the sampling interval. Define two vectors:

T
u(t) : = y T (t) v T (t) ∈ Rnu
(6)

T
û(tk ) : = ŷ T (tk ) v̂ T (tk ) ∈ Rnu ,
with nu := ny + nv . u(t) is the output of the implementation,
û(t) is the input of the implementation. A zero-order hold
mechanism is applied between samplings to the input. At each
sampling time tk , the input applied to the implementation
û(tk ) is updated ∀tk ∈ Ts :
(
u(tk ),
if ku(tk ) − û(tk )k > σku(tk )k
û(tk ) =
(7)
û(tk−1 ), if ku(tk ) − û(tk )k ≤ σku(tk )k,
where σ > 0 is a given constant. Reformulating the event
condition as a quadratic form, the event sequence can be
defined by:

Te := tb |b ∈ N, tb ∈ Ts , ξ T (tb )Qξ(tb ) > 0 .
(8)
 T

T
where ξ(t) := ξp (t) ξcT (t) ŷ T (t) v̂ T (t) ∈ Rnξ , with
nξ := np + nc + ny + nv . And:


Q1 Q2
Q=
QT
Q4
2
in which:



0
(1 − σ)CpT Cp


Q
=
 1

0
(1 − σ)CcT Cc






−CpT
0
Q2 =
(1 − σ)CcT Dc −CcT







I + (1 − σ)DcT Dc −DcT


 Q4 =
−Dc
I
0 is a zero matrix with proper dimension, I is an identity
matrix with appropriate dimension.
It is obvious that Te ⊆ Ts . According to Theorem V.2 in
[13], if the hypothesis therein are satisfied, then the system
(3-8):
1) is globally exponential stable (GES), i.e. ∃c > 0 and ρ >
0 s.t. ∀ξ(0) ∈ Rnξ with w = 0, kξ(t)k ≤ ce−ρt kξ(0)k
for all t ∈ R+ .
2) has L2 -gain from w to z smaller than or equal to γ, i.e.
∃σ : Rnξ → R+ s.t. ∀w ∈ L2 , ξ(0) ∈ Rnξ , the corresponding solution to the system and z(t) := g(ξ(t), w(t))
satisfies kzkL2 ≤ σ(ξ(0)) + γkwkL2 .

To model the timing behaviour of a PETC system, we aim at
constructing a power quotient system for this implementation.
Remark 3.1: Because of the uncertainty brought by the
perturbation, it may happen that the perturbation compensates
the effect of sampling, helping the state of the implementation
to converge. Therefore the event condition in (8) may not be
satisfied along the timeline. As a result, there may not be an
upper bound for the event intervals. However an upper bound
is necessary for constructing a useful power quotient system.
Remark 3.2: To apply scheduling approaches, an online
scheduler is required. The model we are going to construct
is non-deterministic, meaning that after an event the system
may end up in several possible regions, but those regions are
defined in terms of ξp , which means that from a measurement
is not always clear in which region the system is. That means
this online scheduler cannot figure out where the system
is from simple output measurements. Therefore, the online
scheduler should be able to access in which region the system
is.
Assumption 3.3: The current state region at each eventtriggered time tb can be obtained in real time.
Because of the observation in Remark 3.1, we use instead
the following event condition:
tb+1 = inf {tk |tk ∈ Ts , tk > tb ,
o
_
ξ T (tk )Qξ(tk ) > 0 tk ≥ tb + τ̄R(ξ(tb )) ,

(9)

where R(ξ(tb )) is the state region on state-space Rnξ at last
sampling time tb , τ̄R(ξ(tb )) is a regional maximum allowable
event interval (MAEI), which is dependent on R(ξ(tb )).
According to Assumption 3.3, R(ξ(tb )) is obtainable. If
this value is not possible to be accessed by the triggering
mechanisms, one can always employ a global upper bound
τ̄ :≥ τ̄R(ξ(tb )) . We will discuss the computation of τ̄R(tb )
in later sections. Note that, if the PETC implementation
employing (8) can guarantee some pre-designed stability and
performance, then the PETC implementation employing (9)
can guarantee the same stability and performance.
Consider a period:
τ (x) := tb+1 − tb = kx h.

(10)

By definition û(t) is constant ∀t ∈ [tb , tb+1 [ and dependent
on ξp (tb ) and ξc (tb ). The input û(t) can be expressed as:


Cp
0
û(t) = CE x, CE :=
,
Dc Cp Cc
where


T
x := ξpT (tb ) ξcT (tb ) .


T
Let ξx (k) := ξpT (tb + kh) ξcT (tb + kh)
be the state
 T
T
T
evolution with initial state x = ξp (tb ) ξc (tb ) , and k ∈ N.
Now ξx (k) can be computed as:
ξx (k) = M (k)x + Θ(k),
where
M (k) :=





M1 (k)
Θ1 (k)
,
, Θ(k) :=
M2 (k)
0

(11)

4


Z kh







eAp s ds Ap I 0
M1 (k) := I 0 +



0






+Bp Dc Cp Cc ,


X

 k−1


k

0
I
M
(k)
:=A
+
Ak−1−i
Bc Cp 0 ,
2

c
c



i=0


Z kh




 Θ1 (k) :=
eAp (kh−s) Ew(s)ds.

Figure 1. An example of the state space partition, into (a) finite polyhedral
cones, (b) finite homocentric spheres, and (c) finite state-space partition.

0

Thus from the event condition in (9), kx in (10) can be
computed by:

(12)
kx = min k x , k x ,
τ̄

and
where k x := R(x)
h

k x := inf k ∈ N+
))

T 

M (k)x + Θ(k)
M (k)x + Θ(k)
Q
>0
.
CE x
CE x
(13)
Now we present the main problem to be solved in this paper.
Consider the system:
S = (X, X0 , U, −→, Y, H),

(14)

where
n
• X = R x , nx = np + nc ;
n
• X0 ⊆ R x ;
• U = ∅;
0
0
• −→⊆ X × U × X such that ∀x, x ∈ X : (x, x ) ∈−→
0
iff ξx (H(x)) = x ;
+
• Y ⊂N ;
nx
• H :R
→ N+ where H(x) = kx .
S is an infinite-state system. The output set Y of system S
contains all the possible amount of sampling steps tb+1h−tb ∈
N, b ∈ N that the system (3-7), and (9) may exhibit. Once
the sampling time h is chosen, the event interval then can be
computed by kx h.
Problem 3.4: Construct a finite abstraction of system S
capturing enough information for scheduling.
Inspired by [16], we solve this problem by constructing a
power quotient systems S/P based on an adequately designed
equivalence relation P defined over the state set X of S. The
constructed systems S/P are semantically equivalent to timed
automata, which can be used for automatic scheduler design
[15].
In particular, the system S/P to be constructed is as follows:


S/P = X/P , X/P,0 , U/P , −→, Y/P , H/P ,
(15)
/P

•
•
•

X/P = Rn/Px := {R1 , · · · , Rq };
X
= Rn/Px ;
 /P,0

x/P , x0/P ∈−→ if ∃x ∈ x/P , ∃x0 ∈ x0/P such that

Compared with the power quotient system constructed in
[16], a main difference is that since we focus on PETC, there
is no timing uncertainty.
IV. C ONSTRUCTION OF THE QUOTIENT SYSTEM
A. State set
From the results in [8], we remark the following fact:
Remark 4.1: When w = 0, excluding the origin, all the
states that lie on a line going through the origin have an
identical triggering behavior.
We also call the following assumption:
Assumption 4.2: The perturbation w satisfies w ∈ L2 and
w ∈ L∞ . Besides, assume an upper bound W > 0 for kwkL∞ ,
i.e. kwkL∞ ≤ W, is known.
Base on Remark 4.1 and Assumption 4.2, we propose statespace partition as follows:
(
n^
x −1
xT Ξs1 ,(i,i+1) x ≥ 0
Rs1 ,s2 = x ∈ Rnx
(16)
i=1
o
^
Ws2 −1 ≤ |x| < Ws2 ,
where s1 ∈ {1, · · · , q1 }, s2 ∈ {1, · · · , q2 }, q1 , q2 ∈ N
are pre-designed scalars. Ξs1 ,(i,j) is a constructed matrix;
{Wi |i ∈ {0, · · · , q2 }} is a sequence of scalars. Note that
W0 = 0, Wq2 = +∞, and the rest Ws2 are bounded
and
It is obvious that
S somewhere in between 0 and +∞.
nx
R
=
R
.
s1 ,s2
s1 ∈{1,··· ,q1 },s2 ∈{1,··· ,q2 }
This state-space partition combines partitioning the statespace into finite polyhedral cones (named as isotropic covering
[8]) and finite homocentric spheres. From (16), we can see
that, the isotropic covering describes the relation between
entries of the state vector, while the transverse isotropic
covering is used to capture the relation between the norm of
the state vector and the L∞ norm of the perturbations, which
will be shown later in Theorem 4.4. If w = 0, the homocentric
spheres can be omitted.
Details on the isotropic covering can be found in the
Appendix. Figure 1 shows a 2-dimensional example.

/P

•
•

ξx (H(x)) = x0 ;
Y/P ⊂ 2Y ⊂ IN+;

H
h /P x/P i= minx∈x/P H(x), maxx∈x/P H(x) :=
k x/P , k̄x/P .

S/P is a finite state system.

B. Output map
We first free the system dynamics from the uncertainty
brought by the perturbation.
Lemma 4.3: Consider the system (3-7) and (9), and that
Assumptions 4.2 hold. If there exist a scalar µ ≥ 0 and a

5

symmetric matrix Ψ such that (Q1 + Ψ)1  µI, then k x
generated by (13) is lower bounded by:
kx0 := inf{k ∈ N+ |Φ(k)  0},
where
Q1 + Ψ =


(Q1 + Ψ)1
(Q1 + Ψ)3

(Q1 + Ψ)2
(Q1 + Ψ)4

(17)


(Q1 + Ψ)1 ∈ Rnp ×np ,


Φ1 (k) Φ2 (k)
0
−Ψ
0 ,
Φ(k) := ΦT
2 (k)
0
0
Φ3 (k)

T
T

 Φ1 (k) =M (k)Q1 M (k) + M (k)Q2 CE


T
T

+ CE
Q3 M (k) + CE
Q4 CE
T
T

Φ2 (k) =M (k)Q1 + CE Q3




 Φ (k) =khµλ
T
3
max E E dAp (k),

(18)

(19)

and

kλmax (Ap +AT
p) − 1


e
 , if λmax Ap + AT
p 6= 0,
T
dAp (k) =
λmax Ap + Ap



kh,
if λmax Ap + AT
p = 0.
Next we construct LMIs that bridge Lemma 4.3 and the
state-space partition.
Theorem 4.4: (Regional lower bound) Consider a scalar
k s1 ,s2 ∈ N and regions with s2 > 1. If all the hypothesis in
Lemma 4.3 hold and there exist scalars εk,(s1 ,s2 ),(i,i+1) ≥ 0
where i ∈ {1, · · · , nx −1} such that for all k ∈ {0, · · · , k s1 ,s2 }
the following LMIs hold:


H
Φ2 (k)
 0,
(20)
ΦT
−Ψ
2 (k)

according to Remark 4.1, only applying isotropic covering is
enough.
We define Rs1 ,• to represent Rs1 ,s2 , ∀s2 ∈ {1, · · · , q2 }.
Corollary 4.7: (Regional lower bound when w = 0) Consider a scalar k s1 ,• ∈ N. If there exist scalars εk,s1 ,(i,i+1) ≥ 0
where i ∈ {1, · · · , nx −1} such that for all k ∈ {0, · · · , k s1 ,• }
the following LMIs hold:
X
Φ1 (k) +
εk,s1 ,(i,i+1) Ξs1 ,(i,i+1)  0,
(22)
i∈{1,··· ,nx −1}

with Φ1 (k) defined in (19), then the inter event time (8) of the
system (3-7) with w = 0 are regionally bounded from below
by (k s1 ,• + 1)h.
Corollary 4.8: (Regional upper bound when w = 0) Let
¯l ∈ N be a large enough scalar. Consider a scalar k̄s ,• ∈
1

k s1 ,• , · · · , ¯l . If there exist scalars ε̄k,s1 ,(i,i+1)
≥ 0 where

i ∈ {1, · · · , nx − 1} such that for all k ∈ k̄s1 ,• , · · · , ¯l the
following LMIs hold:
X
Φ1 (k) −
ε̄k,s1 ,(i,i+1) Ξs1 ,(i,i+1)  0,
(23)
i∈{1,··· ,nx −1}

with Φ1 (k) defined in (19), then the inter event time (8) of the
system (3-7) with w = 0 are regionally bounded from above
by k̄s1 ,• h.
Remark 4.9: For the choice of ¯l, we follow Remark 2 in [16],
and apply a line search approach: increasing ¯l until Φ1 (¯l)  0.
This results in ¯l being a global upper bound for the inter event
time (8) of the system (3-7) with w = 0.
It is obvious that ¯l ≥ k̄s1 ,• > k s1 ,• ≥ k s1 ,s2 , ∀s2 . We can
now set the regional MAEI τ̄R(ξ(tb )) in (9) as: τ̄R(ξ(tb )) :=
k̄s1 ,• h, ∀x ∈ Rs1 ,• .

where
H =Φ1 (k) + Φ3 (k)W 2 Ws−2
I
2 −1
X
+
εk,(s1 ,s2 ),(i,i+1) Ξs1 ,(i,i+1) ,
i∈{1,··· ,nx −1}

with Φ1 (k), Φ2 (k), and Φ3 (k) defined in (19), and Ψ from
Lemma 4.3, then the inter event times (9) for system (3-7)
are regionally bounded from below by (k s1 ,s2 + 1)h. For the
regions with s2 = 1, the regional lower bound is h.
Remark 4.5: In Theorem 4.4, we discuss the situations when
s2 > 1 and s2 = 1, since for all regions with s2 > 1, it holds
that Ws2 −1 6= 0; while for all regions with s2 = 1, Ws2 −1 = 0
holds. When Ws2 −1 6= 0, one can easily validate the feasibility
of the LMI (20); while when Ws2 −1 = 0, H will be diagonal
infinity, making the LMI (20) infeasible when k > 0. However,
according to the property of PETC, i.e. tb+1 ∈ Ts and tb+1 >
tb , the regional lower bound exists and is equal to h.
Following similar ideas of Theorem 4.4, we present next
lower and upper bounds starting from each state partition when
w = 0. Consider the following event condition:
(
)

T 

M
(k)x
M
(k)x
kx = inf k ∈ N+
Q
>0 .
(21)
CE x
CE x
Remark 4.6: Since (21) does not consider perturbations,
when computing the lower and upper bound for each region,

C. Transition relation
In this subsection, we discuss the construction of the transition relation and the reachable state set. Denote the initial
state set as X0,(s1 ,s2 ) , after k-th samplings without an update,
the reachable state set is denoted as Xk,(s1 ,s2 ) . According to
(11), a relation can be obtained as:
Xk,(s1 ,s2 ) = M (k)X0,(s1 ,s2 ) + Θ(k).

(24)

It is obvious that, Xk,(s1 ,s2 ) cannot be computed directly,
because the perturbation is uncertain and the state region may
not be convex. Therefore, we aim to find sets X̂k,(s1 ,s2 ) such
that:
Xk,(s1 ,s2 ) ⊆ X̂k,(s1 ,s2 ) .
To compute X̂k,(s1 ,s2 ) , we take the following steps:
1) Partition the dynamics: According to (24), X̂k,(s1 ,s2 )
can be computed by:
1
2
X̂k,(s1 ,s2 ) = X̂k,(s
⊕ X̂k,(s
,
1 ,s2 )
1 ,s2 )
1
2
where ⊕ is the Minkowski addition, X̂k,(s
and X̂k,(s
1 ,s2 )
1 ,s2 )
are sets to be computed.

6

1
1
2) Compute X̂k,(s
: One can compute X̂k,(s
by:
1 ,s2 )
1 ,s2 )
1
X̂k,(s
= M (k)X̂0,(s1 ,s2 ) ,
1 ,s2 )

where X̂0,(s1 ,s2 ) is a polytope that over approximates
X0,(s1 ,s2 ) , i.e. X0,(s1 ,s2 ) ⊆ X̂0,(s1 ,s2 ) . X̂0,(s1 ,s2 ) can be
computed as in the optimization problem (1) in [2].
2
2
3) Compute X̂k,(s
: For the computation of X̂k,(s
,
1 ,s2 )
1 ,s2 )
it follows that:
2
= {x ∈ Rnx ||x| ≤ |Θ(k)|},
X̂k,(s
1 ,s2 )

where
kh

Z

eAp (kh−s) Ew(s)ds

|Θ(k)| =
0
kh

Z

eAp (kh−s) Ew(s) ds

≤
0
kh

Z

eAp (kh−s) ds|E|kwkL∞

≤
0

kh λ
max

Z
≤

e



AT
p +Ap
2

Figure 2. State-space partition and the labeling of each region.


(kh−s)

ds|E|W.

0

In which the last inequation holds according to (2.2) in [24].
of the system
Thus the reachable set X{k
s1 ,s2 ,k s1 ,• },(s1 ,s2 )
(3-7), and (9) starting from X0,(s1 ,s2 ) can be computed by:
X{k
⊆ X̂{k
s1 ,s2 ,k s1 ,• },(s1 ,s2 )
s1 ,s2 ,k s1 ,• },(s1 ,s2 )
[
=
X̂k,(s1 ,s2 ) .
k∈{ks ,s ,··· ,ks1 ,• }
1 2
To compute the transitions in S/P , one can check the
intersection between the over approximation of reachable state
set and all the state regions Rs01 ,s02 , ∀s01 ∈ {1, · · · , q1 }, s02 ∈
{1, · · · , q2 }. More specifically, one can check if the following
feasibility problem for each state region holds:
Rs01 ,s02 ∩ X̂{k

s1 ,s2 ,k s1 ,•

},(s1 ,s2 ) 6= ∅,

Figure 3. Computed result of the regional lower bound with W = 2.

in which case

Rs1 ,s2 , Rs01 ,s02 ∈−→ .
/P

D. Main result
Now we summarize the main result of the paper in the
following theorem.
=
 Theorem 4.10: The metric
 system S/P
X/P , X/P,0 , U/P , −→, Y/P , H/P ,
-approximately
/P

simulates S, where  = max dH (y, y 0 ), y = H(x) ∈ Y ,
y 0 = H/P (x0 ) ∈ Y/P , ∀ (x, x0 ) ∈ P , and dH (·, ·) is the
Hausdorff distance.
V. N UMERICAL EXAMPLE
In this section, we consider a system employed in [13] and
[22]. The plant is given by:


 
 
˙ = 0 1 ξ(t) + 0 v(t) + 1 w(t),
ξ(t)
−2 3
1
0

and the controller is given by:

K= 1


−4 .

This plant is chosen since it is easy to show the feasibility of
the presented theory in 2-dimensional plots. The state-space
partition is presented in Figure 2.
We set W = 2, the convergence rate ρ = 0.01, L2 gain
γ = 2, sampling time h = 0.005s, event condition σ = 0.1.
By checking the LMI presented in [13], we can see there
exists a feasible solution, thus the stability and performance
can be guaranteed. The result of the computed lower bound by
Theorem 4.4 is shown in Figure 3. Figure 4 shows a zoomed-in
version. The computed upper bound by Corollary 4.8 is shown
in Figure 5. The resulting abstraction precision is  = 0.15s.
The simulation results of the system evolution and event
intervals with perturbations are shown in Figure 6. The upper
bound triggered 6 events during the 10s simulation. Note
that, increasing the number of subdivisions can lead to less

7

Figure 4. Zoomed-in result of the regional lower bound with W = 2.
Figure 6. System evolution and event intervals when w = 2 sin(πt), t ∈
[3, 8]: state evaluation and perturbance, event intervals with the bounds.

Figure 5. Computed result of the regional upper bound with w = 0.

conserved lower and upper bounds of the inter event time.
The conservativeness can also be reduced by decreasing W.
The reachable state regions starting from each region is
shown in Figure 7. As an example, the reachable state region
of the initial region (s1 , s2 ) = (4, 6) is shown in Figure 8.
We also present a simulation when w = 0. The lower
bound is shown in Figure 9. The evolution of the system is
shown in Figure 10, which shows that, the inter event intervals
are within the computed bounds. The reachable state regions
starting from each region are shown in Figure 11.
VI. C ONCLUSION
In this paper, we present a construction of a power quotient
system for the traffic model of the PETC implementations
from [13]. The constructed models can be used to estimate the
next event time and the state set when the next event occurs.
These models allow to design scheduling to improve listening
time of wireless communications and medium access time to
increase the energy consumption and bandwidth occupation
efficiency.

Figure 7. Reachable regions starting from each state region, with labeling
from Figure 2.

In this paper, we consider an output feedback system with
a dynamic controller. However, the state partition is still
based on the states of the system and controller. The system
state may not always be obtainable. Therefore, to estimate
the system state in an ETC implementation from output
measurements is a very important extension to make this work
more practical. The periodic asynchronous event-triggered
control (PAETC) presented in [10] is an extension of PETC
considering quantization. One can either treat the quantization
error as part of the perturbations, or analyze this part separately
to increase the abstraction precision, since the dynamics of the
quantization error is dependent on the states. This is also an
interesting future investigation. Another interesting extension
is reconstruction of traffic models for each sensor node to
capture the local timing behavior in a decentralized PETC

8

Figure 8. Flow pipe of (s1 , s2 ) = (4, 6): indicating initial state set (red),
reachable state set (blue), and reachable regions (cyan).

Figure 10. System evolution and event intervals when w = 0: state evaluation
and event intervals vs computed bounds.

Figure 9. Computed result of the regional lower bound with w = 0.

implementation, by either global information or even only
local information.
A PPENDIX
Isotropic
covering:
Consider
x
=

T
x1 x2 · · · xn
∈ Rn . We first present a case
when x ∈ R2 . Let Θ = [− π2 , π2 [ be an interval. Splitting this
interval into q sub-intervals and Θs = [θs , θs [ be the s-th
sub-interval. Then for each sub-interval, one can construct a
cone pointing at the origin:
n
o
Rs = x ∈ R2 |xT Ξ̃s x ≥ 0 ,
where

Ξ̃s =


− sin θs sin θs 12 sin(θs + θs )
.
1
2 sin(θ s + θ s ) − cos θ s cos θ s

Remark 4.1 shows that x and −x have the same behaviours,
therefore it is sufficient to only consider half of the state-space.

Figure 11. Reachable regions starting from each conic region, with labeling
from Figure 2.

Now we derive the case when x ∈ Rn , n > 2. Define
(x)i,j = (xi , xj ) as the projection of this point on its i − j
coordinate. Now a polyhedral cone Rs can be defined as:
(
)
n−1
^
n
T
Rs = x ∈ R
(x)(i,i+1) Ξ̃s,(i,i+1) (x)(i,i+1) ≥ 0 ,
i=1

where Ξ̃s,(i,i+1) is a constructed matrix. A relation between
Ξ̃s,(i,i+1) and Ξs,(i,i+1) from (16) is given by:

[Ξs,(i,i+1) ](i,i)
=[Ξ̃s,(i,i+1) ](1,1)





=[Ξ̃s,(i,i+1) ](1,2)

 [Ξs,(i,i+1) ](i,i+1)
[Ξs,(i,i+1) ](i+1,i)




[Ξs,(i,i+1) ](i+1,i+1)



[Ξs,(i,i+1) ](k,l)

=[Ξ̃s,(i,i+1) ](2,1)

=[Ξ̃s,(i,i+1) ](2,2)
=0,

9

where [M ](i,j) is the i-th row, j-th column entry of the matrix
M , k and l satisfy (k, l) 6= (i, i + 1).
Proof of Lemma 4.3: We decouple the event triggering
mechanism in (13) first:

T 

M (k)x + Θ(k)
M (k)x + Θ(k)
Q
CE x
CE x
T

T

=x Φ1 (k)x + x Φ2 (k)Θ(k) + Θ

T

(k)ΦT
2 (k)x

+ ΘT (k)Q1 Θ(k)
T
≤xT (Φ1 (k) + Φ2 (k)Ψ−1 ΦT
2 (k))x + Θ (k)(Q1 + Ψ)Θ(k),
(25)
where the last inequality comes from Lemma 6.2 in [12]. Now
for the uncertainty part, we have:

ΘT (k)(Q1 + Ψ)Θ(k)

T 
Θ1 (k)
(Q1 + Ψ)1
=
0
(Q1 + Ψ)3

(Q1 + Ψ)2
(Q1 + Ψ)4

Proof of Theorem 4.4: We first consider the regions with
s2 > 1. If all the hypothesis of the theorem hold, by applying
the Schur complement to (20), one has:

(29)
xT H + Φ2 (k)Ψ−1 ΦT
2 (k) x ≤ 0.
From (16), and applying the S-procedure, it holds that:

xT Φ1 (k) + Φ3 (k)W 2 Ws−2
I + Φ2 (k)Ψ−1 ΦT
2 (k) x ≤ 0.
2 −1
(30)
From (16) we also have:
xT x ≥ Ws22 −1 .

(31)

Since Φ3 (k), W, and Ws2 −1 are non-negative scalars and
Ws2 −1 > 0, we have the following inequality:
xT Φ3 (k)W 2 Ws−2
Ix = Φ3 (k)W 2 Ws−2
xT x
2 −1
2 −1



Θ1 (k)
0

=ΘT
1 (k)(Q1 + Ψ)1 Θ1 (k).
From the hypothesis of the theorem that there exists µ such
that (Q1 + Ψ)1  µI, together with Jensen’s inequality [12],
inequality (2.2) in [24], and Assumption 4.2, i.e. w ∈ L∞ ,
ΘT (k)(Q1 + Ψ)Θ(k) can be bounded from above by:

≥Φ3 (k)W 2 Ws−2
Ws22 −1 = Φ3 (k)W 2 ≥ Φ3 (k)kwk2L∞ ,
2 −1
(32)
in which the last inequality comes form the definition of W.
Now inserting (32) into (30) results in:

2
xT Φ1 (k) + Φ2 (k)Ψ−1 ΦT
2 (k) x + Φ3 (k)kwkL∞ ≤ 0,

which together with applying the Schur complement to (18)
provides the regional lower bound.
ΘT (k)(Q1 + Ψ)Θ(k) = ΘT
1 (k)(Q1 + Ψ)1 Θ1 (k)
!
!
When s2 = 1, k > 0, H will be diagonal infinity. Thus the
T Z
Z kh
kh
LMI (20) will be infeasible. According to the event-triggered
≤µ
eAp (kh−s) Ew(s)ds
eAp (kh−s) Ew(s)ds
condition (9), which indicates that tb+1 ∈ Ts and tb+1 > tb ,
0
0
the regional lower bound for those regions with s2 = 1 is h.
(by (Q1 + Ψ)  µI)
This finishes the proof.
Z kh 
T 

Proof of Corollary 4.7: The result can be easily obtained
≤khµ
eAp (kh−s) Ew(s)
eAp (kh−s) Ew(s) ds
0
from Theorem 4.4 considering E = 0.
(by Jensen’s equality)
Proof of Corollary 4.8: The result can be easily obtained
Z kh
analogously
to Theorem 4.4 considering E = 0: if all the
T
≤khµ
e(kh−s)λmax (Ap +Ap ) wT (s)E T Ew(s)ds
hypothesis of this
 Corollary hold, then according to (23),
0
Φ
(k)

0,
k
∈
k̄s1 ,• , · · · , ¯l . According to the definition of
1
(by (2.2) in [24])
Φ1 (k) in (19), for all k ≥ k̄s1 ,• , it holds that:
Z
 kh (kh−s)λmax (Ap +AT )
p dskwk2

T 

≤khµλmax E T E
e
L∞
M (k)x
M (k)x
0
Q
> 0,
CE x
CE x
(by w ∈ L∞ )

=khµλmax E T E dAp (k)kwk2L∞ .
which together with event condition (21) provides the regional
(26) upper bound.
With (26), (25) can be further bounded as:
Proof of Theorem 4.10: The result follows from Lemma

T 

2.7
and
the construction described in this section.
M (k)x + Θ(k)
M (k)x + Θ(k)
Q
CE x
CE x
(27)

R EFERENCES
T
−1 T
2
≤x Φ1 (k) + Φ2 (k)Ψ Φ2 (k) x + Φ3 (k)kwkL∞ .
From the hypothesis of the theorem, if Φ(k)  0 holds,
then by applying the Schur complement to (18), the following
inequality holds:

2
xT Φ1 (k) + Φ2 (k)Ψ−1 ΦT
2 (k) x + Φ3 (k)kwkL∞ ≤ 0,
which indicates:

T 

M (k)x + Θ(k)
M (k)x + Θ(k)
Q
≤ 0.
CE x
CE x

(28)

Therefore, k x generated by (13) is lower bounded by kx0
generated by (17). This ends the proof.

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